ruif/phdthesis/chapter 4_x.pdf
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CHAPTER 4
ANALYSIS OF THE GOVERNING EQUATIONS
4.1 INTRODUCTION
The mathematical nature of the systems of governing equations deduced in Chapters 2 and 3
is investigated in this chapter. The systems of PDEs that express the quasi-equilibrium
approximation are studied in greater depth. The analysis is especially important for the
systems whose solutions are likely to feature discontinuities, as a result of strong gradients
growing steeper, or because the initial data is already discontinuous. The geomorphic
shallow-water flows, such as the dam-break flow considered in Chapter 3, are the
paradigmatic example of flows for which discontinuities arise fundamentally because of the
initial conditions.
Laboratory experiments show that, in the first instants of a sudden collapse of a dam, vertical
accelerations are strong and a bore is formed, either through the breaking of a wave (Stansby
et al. 1998) or due to the incorporation of bed material (Capart 2000, Leal et al. 2002).
Intense erosion occurs in the vicinity of the dam and a highly saturated wave front is likely to
form at 0 4t t t≡ ≈ , 0 0t h g= , where h0 is the initial water depth in the reservoir. The
saturated wave front can be seen forming in figure 3.1(a). Unlike the debris flow resulting
from avalanches or lahars, the saturated front is followed by a sheet-flow similar to that
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encountered in surf or swash zones (Asano 1995), as seen in figures 3.2(b) and (c). The
intensity of the sediment transport decreases in the upstream direction as the flow velocities
approach fluvial values. While the flow is highly erosive in the wave front region, sediment
debulking may result into generalised deposition as the flow velocity decreases. Thus, the
solution of the competent system of equations comprises continuous reaches eventually
separated by discontinuities.
If the collapse of a dam should be idealised as an instantaneous removal of a vertical barrier,
initially separating two constant states that extend indefinitely on both up- and downstream
directions, as seen in figure 4.1, the dam-break problem is a Riemann problem. Riemann
problems admit self-similar solutions relatively to the variable 0x t gh if the hyperbolic
equations are homogeneous, i.e., if G = 0. For special cases of the flux vector, F, explicit
expressions for the dependent variables, functions of time and spatial co-ordinates, are
attainable, as it is the case of the flat- fixed-bed solution for the shallow water equations
presented by Ritter in 1887. The latter is generalized and thoroughly described by Stoker
(1958), pp. 311-326, 333-341 and 513-522.
The importance of explicit theoretical solutions is threefold: i) they are computationally
simpler than numerical solutions, ii) they provide an order of magnitude and important
phenomenological insights on the behaviour of the system under more general conditions and
iii) they provide a way to access the quality of numerical discretization techniques. Stoker’s
or Ritter’s solutions have been used to verify the quality of, virtually, all numerical models
build ever since Stoker’s reference book was published. They also provided an elemental
proof of existence of a weak solution for the shallow water equations and sparkled important
theoretical advances on the hydrodynamics of unsteady open-channel flow (e.g. Su & Barnes
1970, Hunt 1983, 1984). The practical use of Stoker’s solution was extended to play the role
of a reference situation for the interpretation of experimental results on dam-break flows. As
an example, Ritter’s value for the velocity of the dam-break wave front, 02 gh , is the
reference order of magnitude of the dam-break flood wave propagation, to which every
experimental result is compared and at whose light is discussed.
Pure hydrodynamic models, Stoker’s solution included, fail to reproduce the characteristic
time and length scales of the dam-break flow when morphological impacts are important.
Because of this fundamental inadequacy, research in geomorphic dam-break flows has been
conducted through a combination of fieldwork, laboratory physical modelling, theoretical
analysis and numerical simulation. Research projects like CADAM and IMPACT provided the
framework for a number of studies, notably Capart & Young (1998), Fraccarollo & Capart
(2002) or Leal et al. (2005), that resulted in major advances, comparable to those
proportioned by the landmark works of Dressler (1952, 1954), Whitham (1955) and Stoker,
op. cit., in the conceptualisation of the phenomena involved and in the development of
simulation capabilities.
Especially relevant is the study of Fraccarollo & Capart (2002) whom, in the wake of works
by Capart & Young (1998) and Fraccarollo & Armanini (1999), have built a solution for the
Riemann problem posed by the homogeneous geomorphic shallow-water equations subjected
to initial conditions comprising a jump in the water level. The solution is not explicit because
of the strong non-linearity of the closure equations. Numerical computations are unavoidable
because the invariants of the simple centred waves can not be explicitly found.
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FIGURE 4.1. Graphic depiction of the initial conditions for the Riemann problem posed to the
geomorphic shallow water equations. The variables, Y, R, and Yb, are, respectively, the
water level, the unit mass discharge and the bed elevation. The subscripts L and R
stand, respectively, for the left and right states, respectively.
The main objectives of this chapter are, in the wake of Fraccarollo & Capart (2002), the
development of a weak solution of the Riemann problem for the geomorphic shallow water
equations and the description of the main features of the wave structure. Special attention will
be devoted to the condition of existence of alternative wave structures, depending on the
initial data. The initial values for the Cauchy problem are the left and right states,
characterised by the water elevation, Y, bed elevation, Yb, and total mass discharge per unit
width, R. Adding to the discontinuity in the water level, the initial discontinuity in the bed
elevation is, thus, explicitly addressed.
Most of the chapter is dedicated to the study of the characteristic fields for which
discontinuities are likely to develop. It is necessary to investigate the fundamental properties
of the characteristic fields, notably signal, monotonicity and non-linearity. In addition,
because the debate on the role of non-linear algebraic equations, describing important
physical phenomena such as sediment transport capacity or bulk flow resistance, is yet to be
closed, questions concerning existence and uniqueness of the solution must, thus, be posed.
The existence of the solutions for the Riemann problem was proved by Lax (1957) for a finite
set of strictly hyperbolic, genuinely non-linear, conservation equations. However, the proof is
valid for the case of small discontinuities in the initial data, which is generally not the case in
the dam-break problem. Further works by Glimm (1965), the first proof for arbitrary initial
values, Smoller (1969), di Perna (1973), Dafermos (1973) or Liu (1974) helped building a
library of theoretical results that may be used as a guide to establish the conditions of
existence and uniqueness of the Riemann solution of the geomorphic shallow water equations.
Considerations on the existence and unicity of the solution of the Riemann problem for the
geomorphic shallow water equations are, thus, legitimate and will be addressed.
The text is structured so as to highlight the main objectives mentioned above. Wave-like
description of wave forms and hyperbolicity are discussed in §4.2.1. The governing equations
YbL YbR
YL
YR
y/h0
x/h0
RL
RR
x0/h00
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subjected to analysis are described in §4.2.2, namely in what concerns the embedded
hypotheses. The following sections, §4.2.3 to §4.2.5 are dedicated to the mathematical
analysis of the system of equations. Special attention is given to the type of hyperbolicity and
non-linearity of the system of equations. The properties of each of the characteristic fields
are investigated, notably signal, monotonicity and non-linearity.
Two possible Riemann wave structures are identified in §4.3. The conditions for the existence
of each of the types of solution is discussed in §4.3.2. Entropy-compatible solutions are
calculated in §4.4, with shocks determined by the Rankine-Hugoniot jump equations. The
existence of Riemann invariants for the simple waves encountered is also discussed.
4.2 MATHEMATICAL ANALYSIS OF THE CHARACTERISTIC FIELDS
4.2.1 Notes on hyperbolicity and non-linear propagation of non-linear hyperbolic waves
4.2.1.1 Source terms and hyperbolicity
The generic quasi-linear, autonomous, non-conservative form of the governing equations is
( ) ( )it i x∂ + ∂ =V V GA B (4.1)
where A e Bi, i = 1…m, are real bounded matrix-valued functions of the dependent
variables, m is the dimension of the number of space-like variables, V is the n-dimensional
vector of dependent variables and G, the source term, is a n-dimensional vector valued
bounded function of the dependent variables.
The source terms are of paramount importance in what concerns the quality of the solutions,
understood as its physical plausibility and agreement with observations. They are less
important for the study of the mathematical properties of the system, a claim that is
substantiated next. It should be made clear that the study of the mathematical properties are,
in this chapter, restricted to the qualitative discussion of the solutions, namely existence and
unicity, and to the study of the nature of the propagation of information, in particular type and
number of conditions at the contour of the solution domain.
Regarding the existence and unicity of the solutions, if the initial conditions are smooth
bounded functions and the components of the matrixes A to Bm are smooth functions of the
dependent variables, there is a region around the initial conditions in which the solution exists
and is unique provided that the source term, G, is integrable. If the initial conditions are less
well behaved, existence and unicity are difficult to establish (Dafermos 2000, p. 50) but the
necessary condition concerning G is still its integrability. As seen below, it is easily shown
that the source terms devised in Chapters 2 and 3 do posses the regularity requirements that
warrant its integrability.
As for the nature of the propagation of the information inside the solution domain, the
fundamental propagation typologies are the hyperbolic, the elliptic, the parabolic and the
respective hybrids. The source terms are not fundamental for the filiation of (4.1) in the later
categories. Support for this claim can be found in the elegant account of non-linear wave-like
propagations of Jeffrey & Taniuty (1964), p. 3-9. In particular, the nature of the source terms
is irrelevant for the definition of the type and number of boundary and initial conditions. Thus,
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in the remainder of this chapter, the source terms G of (4.1) will be discarded and the
homogeneous system
( ) ( )it i x∂ + ∂ =V V 0A B (4.2)
will be investigated.
4.2.1.2 Wave-like description of wave forms
The solution of (4.2) is sought as a combination of wave forms. A wave form is imagined as a
bounded piece-wise continuous vector valued function of the space and time co-ordinates
that is superimposed to an equilibrium state. These regularity properties are enough to allow
for a Fourier description of the wave form and hence, without loss of generality, it is assumed
that the wave form is obtained by linear superposition of an infinite number of harmonic
waves. The nature of the propagation of the wave form is discussed next.
The idea underlying the search for solutions with a wave-like behaviour for the quasi-linear
system (4.2) is to take advantage of some of the well known properties of the linear
propagation of periodic waves. For instance, it is known that the Cauchy problem for the
simple advection equation, ( ) ( ) 0t xv v∂ + λ∂ = , where λ is a real constant, admits the solution
( , ) ( )v x t g x t= − λ when the initial condition is { }i( ,0) ( ) Re ( )e xv x g x A x κ= = , where κ is
real and A(x) is piecewise continuous.
Figure 4.2 shows an example where g(x) is defined as above, where A(x) is a real smooth
function with compact support. It is not important that the initial condition is not periodic as
long as it can be extended to display periodicity. As seen in figure 4.2, A(x) was chosen to be
zero outside the interval [ ]0,b , 2b = π κ . A periodic function can be obtained by repeating
A(x) in the intervals ( )1 2 , 2j j− π κ π κ⎡ ⎤⎣ ⎦ , for all integer j.
FIGURE 4.2. Solution of the scalar linear advection equation for the Cauchy problem
{ }i( ,0) Re ( )e xv x A x κ= where 0( ) CA x ∞∈ .
x
t
v(x,t)
dt(X) = λa b
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As shown in figure 4.2, the solution of the simple advection problem in the space-time domain
is simply the displacement, over a distance equal to λt, of the profile exhibited at t = 0. The
amplitude v(x,0) corresponding to each point in the line t = 0 is conveyed, unaltered, along a
line whose slope in the space-time domain is dx/dt = λ.
This result is easy to obtain if it is noticed that each value of ( )0 ,0v x is associated to a
value of 0kx , [ ]0 0, 2x k∈ π . Similarly, at a given value of t, to each value of ( )1 ,v x t t− λ
corresponds a value of 1( )k x t− λ , [ ]1 , 2x t k t∈ λ π + λ . Equation ( )d ( )t X t = λ (figure 4.2)
can also be written ( )k x t K− λ = , which implies 0 1( )kx k x t K= − λ = if 1 0x x t= + λ .
Finally, if 0 1( )kx k x t= − λ then ( ) ( )0 1,0 ,v x v x t t= − λ for all [ ]0 0, 2x k∈ π , t > 0 and
1 0x x t= + λ . Thus, the solution at all times is easily obtained if it is kept in mind that v is
constant along the line, called the phase of the wave, ( ), ( )x t k x t KΣ = − λ = .
It would be important to understand to what extent the procedures valid for the linear solution
can be of use in more complex situations. In this text, while looking for the solution for the
initial-boundary value problem for quasi-linear systems of physically meaningful PDEs, it is
of considerable interest to find variables and coordinates for which initial profiles are purely
advected, by which it is meant unaffected by diffusion or attenuation. This quest leads to the
notion of hyperbolicity. It will be seen that these variables and coordinates are possible to be
found only if the system (4.2) is hyperbolic.
Further discussion of hyperbolicity requires the formal identification of wave form sought as a
solution for (4.2). It is a n-dimensional vector, n being the number of dependent variables,
that can be written as
( )
( ) i( ( ) )( , ) ( ) dep
p tt+∞
ω +
−∞
= ∫ k k xV x V k ki (4.3)
where the components of x are the space co-ordinates, t is the time, the components of k are
the wave numbers of the elemental harmonics in each of the space directions, ( )ω k is the
angular frequency of each of the elemental harmonics, i is the imaginary unit and ( )V k are
the weighting factor of each harmonic.
It should be made clear that in a m-dimensional space, there are n vectors kj, j = 1…n, which
are m-dimensional. Each of these corresponds to each of the n entries of the wave form
vector. They are all linearly dependent, i.e., j j k= ςk e where jς are constants and ke is the
direction of propagation of the wave form. The phase of an harmonic corresponding to the
wave number kj, j = 1…n, is ( )j jω +k k xi , where ( )jω k is the angular frequency. It should
also be stressed that the harmonics are weighted in phase and amplitude by ( )j jV k . For the
sake of simplicity, (4.3) is written for the case where the wave number k is the same for all
of the n components of the vector of the dependent variables, i.e., j iς = ς for i and j. This
restriction does not pose limitations to the following developments and can be lifted whenever
necessary.
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If the problem was linear, each of the harmonics would be propagated at constant speed and
the solution would be retrieved by superimposing the displaced harmonics at the desired time.
In quasi-linear problems this procedure is not feasible. Yet, it is possible to write the wave
form in a way that resembles the solutions of the linear advection equation. As seen in Annex
4.1, equation (4.3) can be written as
( )
( ) ( ) i ( , )0( , ) ( , )e
pp p tt t Σ= xV x V x (4.4)
provided that the concept of phase, represented by Σ, is generalised. As is the case for linear
problems, it may be imagined that each ( )( ,0)pV x in the surface t = 0 propagates along lines
of constant phase. In an autonomous system such as (4.2), each value of the wave form at the
origin of the time is propagated with a unique speed. More precisely, at small times near the
origin, there is an injective continuous application that maps a propagation speed to each
( , )tV x . Unlike linear waves, though, the wave form may endure deformation because its
points do not necessarily propagate with the same speed. In this case, the function ( )0
pV in
(4.4) must be a function of the space and time coordinates.
It is now assumed that the application that maps propagation speeds and ( , )tV x exists for
larger times. The method of constant phase is based on this assumption. The propagation of
the wave form is observed as the evolution of the locus of the points with the same phase,
which can be written as
( ) ( )( , )p pt t KΣ = ϖ + =x κ xi (4.5)
In (4.5) K is a constant, κ is a wave number and ϖ is the angular frequency corresponding to
κ. The wave number should be close to that corresponding to the main harmonic contribution
in (4.3) and can be computed as shown in Annex 4.1.
4.2.1.3 Constant phase
In a m-dimensional metric space, the locus of the points whose phase is K is a (m−1)-dimensional manifold. For instance, in
3, the locus of the points with the same phase would
be represented by two-dimensional manifold as seen in figure 4.3. The progression of a three
dimensional wave in the direction of κ is represented by the position of the two-dimensional
manifold Σ(x,t) − K = 0 in two distinct instants, dt apart.
If the most distinctive feature of the wave form is a sharp gradient following the crest (see
figure 3.1, p. 182). It is generally called a wave front. Without loss of generality, and with
considerable gain of visual suggestion, a surface such as that represented in figure 4.3 may
be thought to represent the propagation of a wave front.
In a two-dimensional space, the locus of a given constant phase is a line. The progression of
a two-dimensional wave front can be observed in figure 4.4. The two-dimensional manifold
shown in figure 4.4 represents the locus, in the space-time domain, of the wave front, i.e., its
successive positions over time.
A simpler case, though less easy to depict graphically, is the propagation of a one-
dimensional wave. In a one-dimensional domain the position of the wave front is represented
by a point and its direction is the x direction, the only spatial co-ordinate. The wave
progression, in the space time domain, is represented by a line. Figure 4.5 shows such a line;
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the wave front is, following the constant phase method, represented by the phase isoline
( , ) 0x t KΣ − = .
FIGURE 4.3. Propagation, in the direction of the wave number κ, of a three-dimensional wave
form in 3.
FIGURE 4.4. Propagation of a wave form in a two-dimensional metric space. The locus of the
successive points of the wave front is a two-dimensional manifold in 2 +× .
x1
x2
x3
κ
x2
t
x1
κ
t = t0 t = t0 + dt t = t0 + 2dt
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For a one-dimensional wave, (4.5) is simplified to
( , )x t t x KΣ = ϖ + κ = (4.6)
where the symbols maintain their previous definitions. Again, the phase can be interpreted as
a potential and, along one of its isolines, one has
( ) ( )d d d 0t xt xΣ = ∂ Σ + ∂ Σ = (4.7)
Rearranging the terms of (4.7), it becomes
( )( )
dd
t
x
xt
∂ Σ ϖ= = −
∂ Σ κ (4.8)
Let (4.6) be written as function of a parameter s, so that ( )x X s= , :X + → and
( )t T s= , :T + +→ are continuously differentiable mappings. A vector ( )sc can be
defined as [ ]( ) ( )X s T s=c .
FIGURE 4.5. Propagation of a wave form in a one-dimensional domain.
The derivative of c with respect to s is the tangent of the line of constant phase and is
defined as
( ) ( ) ( ) ( ) ( )d d ds x s t sX T= ∂ + ∂c c c ⇔
( ) ( ) ( )d d ds x s t sX T= +c e e (4.9)
Without loss of generality, let s t= . In that case, attending to (4.8) and to the fact that
( )d d dt X x t= , one has
( ) ( ) ( )ds x t≡ −κ = ϖ + −κC c e e (4.10)
The vector C, depicted in figure 4.5, is, following its definition, tangent to ( , ) 0x t KΣ − = .
The direction of C is, from (4.8) and (4.10), normal to a vector defined as
( ) ( )* x x t t x t= ∂ Σ + ∂ Σ = κ + ϖn e e e e .
x
t
*nC
−κ
ϖ( )x∂ Σ
( )t∂ Σ
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Thus, bearing (4.10) in mind, equation (4.8) expresses the result that the gradient of a
potential is perpendicular to its isolines. The physical meaning of the construction, seen in
4.5, is that any disturbance associated with a particular phase, Σ, is carried, in the space-time
domain, along the respective isoline with a velocity, called phase velocity, equal to ( )ds c . By
disturbance it is meant any superimposition to a state of equilibrium, in accordance to the
notion of wave form (see also Whitham 1974, p. 127).
Re-writing (4.9), the phase velocity is written
( )ds x t x tϖ
= − + = λ +κ
c e e e e (4.11)
where λ is the slope of the direction of Σ, since
( )d d ( )d tx X tt
ϖ= ≡ λ = −
κ (4.12)
The role of λ is fundamental in the study of the qualitative behaviour of the solution of PDEs
and also for its quantification. Given a point P in the solution domain, it is important to know
how many independent lines of constant phase cross that point (the value of p in (4.4)), how
fast will the information propagate along these lines, i.e., how large is λ corresponding to the
pth wave, and what is the nature of the information carried along such lines.
4.2.1.4 Classification of systems of PDEs
The number of independent propagation directions that exists for a system of PDEs
describing physical phenomena is investigated next. The analysis is restricted to one
dimensional systems of more than one dependent variable (m = 1, n ≥ 2), i.e., the governing
equations are in the form of
( ) ( )t x∂ + ∂ =V V 0A B (4.13)
and the wave form solutions are
( )
( ) ( ) i ( , )0 ( , )e
pp p x tx t Σ=V V (4.14)
Equation (4.13) is promptly obtained from (4.2) by setting B1 ≡ B. Introducing (4.6) in the
wave-form solution (4.14) and the latter in (4.13), one has
( ) ( ){ } ( ) ( ){ }( ) ( ) ( ) ( )i( ) i( ) i( ) i( )0 0 0 0e e e ep p p pt x t x t x t x
t t x xϖ +κ ϖ +κ ϖ +κ ϖ +κ∂ + ∂ + ∂ + ∂ =V V V V 0A B
( ){ } ( ){ }( ) ( ) ( ) ( )i( ) i( )0 0 0 0i ie ep p p pt x t x
t xϖ +κ ϖ +κ∂ + ϖ + ∂ + κ =V V V V 0A B (4.15)
the term i( )e t xϖ +κ
can be factored out. It does not constitute a solution because it is different
from zero for all finite real values of the phase. Equation (4.15) becomes
( ) ( ){ } { }( ) ( ) ( )0 0 0
0
ip p pt x
=
∂ + ∂ + κ + ϖ =V V V 0A B B A (4.16)
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As explained above, solutions of (4.13) are sought as wave-forms, written as (4.14). If (4.14)
is a solution of (4.13), it is so for all values of the phase. Thus, if the constant K in (4.6) is
zero, (4.14) reduces to ( ) ( )
0( , ) ( , )p px t x t=V V and (4.13) becomes ( ) ( )( ) ( )0 0
p pt x∂ + ∂ =V V 0A B .
This justifies the elimination of the first two terms in (4.16). The remainder of equation (4.16)
can be written as
( )( ) ( )0
p pκ + ϖ =V 0B A
and, since ( ) ( )p pλ = −ϖ κ (equation (4.12))
( )( ) ( )0
p p− λ =V 0B A (4.17)
Equation (4.17) states that non-trivial solutions ( )0
pV can be found provided that the matrix
− λB A is singular. Thus, the computation of the direction of the phase is a eigenvalue
problem. The condition of singularity of − λB A is expressed by the condition of zero
determinant
( )det 0− λ =B A (4.18)
Equation (4.18) is the characteristic polynomial of 1−A B , admitting that A is non-singular1.
The order of the polynomial is equal to the rank of 1−A B or, equivalently, to the number, n,
of dependent variables. The eigenvalues of 1−A B are the p ≤ n distinct roots of the
characteristic polynomial, simply called characteristics. From (4.11) and (4.12) and from
figure 4.5 it is clear that they are the directions of the lines of constant phase. The phase
velocities of system (4.13) are fully determined by the characteristics of 1−A B . As a result,
the term characteristic, denoted λ, is taken as a substitute of phase velocity when referring to
the direction of a given line of constant phase.
The number and the properties of the characteristics, i.e., the roots of (4.18), determine the
mathematical properties of (4.13) in what concerns information transfer and well-posedness
of initial and boundary conditions, i.e., type and number of boundary and initial conditions.
4.2.1.5 Domains of dependence and influence and conditions at the contour
If all the roots of (4.18) are complex, system (4.13) expresses a diffusive phenomenon. In that
case, system (4.13) is said to be elliptic. If some of the eigenvalues are complex and some
real, the system is said to be hybrid. If the rank of − λB A is odd, and there are complex
roots, the system is necessarily hybrid because the complex roots are conjugate pairs.
Most notably, if all the eigenvalues of 1−A B are complex, then there are no real
characteristic lines. If there are no characteristic lines in the space time domain, the
information cannot be propagated from the contour to the solution domain along lines of
constant phase such as that shown in figure 4.5. The value of V(x,t) at a given point in the
1 If A is singular, the roots of the characteristic polynomial are still propagation paths. Because the
number of roots is less than the rank of A, the solution exhibits a parabolic behaviour (see the
discussion in the next page). Ponce & Simmons (1977) discuss, in the context of the shallow water
equations, the physical consequence of the absence of the time derivatives that, in some coordinate
frame, make A a singular matrix. In the following text it will be assumed that A is non-singular.
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solution domain H (see figure 4.6) cannot be tracked to any specific region in H or any
specific point in ∂H (figure 4.6). Thus, solutions in the form of (4.14) are not possible for
elliptical problems.
Equation (4.14) can, nevertheless, be used to intuit the type of solution obtainable for elliptic
problems. If the eigenvalues of 1−A B are complex, then, by (4.12), so are the angular
frequencies. Consequently, from (4.6) and (4.14) the wave form would be written
( )( ) ( ) i Re( )Im( )
0( , ) ( , )e ep p x ttx t x t κ + ϖ− ϖ=V V (4.19)
The effect of Im( )e t− ϖ
in (4.19) is that of attenuation of the wave amplitude as the time
increases, hence the diffusive character of the elliptic solutions.
Having no definite directions of propagation, system (4.13) admits, in each point P = (xi,ti) belonging to H (figure 4.6), a solution V(P) that depends potentially on all other values of
V(x,t), (x,t) ∈ H. Thus, the domain of dependence of P is the entire domain H. Reciprocally,
the value of V in P may affect the solution on all other points of H, i.e., the domain of
influence of P is also the whole solution domain. This feature of elliptic systems is depicted in
figure 4.6.
FIGURE 4.6. Elliptic systems; domains of influence and of dependence of a given point P. The
solution domain is enclosed by the dashed line .
Obviously, the variable t is not time-like. Inexistence of definite paths for information transfer
implies physical reversibility, incompatible with a time-like behaviour. Thus, in the contour
∂Η (figure 4.6) of the solution domain, it is necessary to provide information where it is
physically relevant. Because there are no time-like variables, the Cauchy problem does not
make sense for elliptic problems.
The solution of the homogeneous problem (4.13) is non-diffusive iff the eigenvalues of (4.18)
are real numbers. Non-diffusive phenomena are related to propagation problems in the broad
sense. If the number of roots of (4.18) is p < n, where n is the rank of the matrix − λB A and
all the roots are real, the system is said to be parabolic. In that case, the algebraic multiplicity
of some of the eigenvalues ( )pλ is larger than one. Figure 4.7 shows an idealised situation,
with n = 4, where the algebraic multiplicity of the two roots is equal to two.
There are less independent directions of propagation than dependent variables. In general,
the information carried by each characteristic line concerns all n dependent variables. Let it
x
t
domain of influence and
of dependence
P Η
∂ΗVj(x,t1), j ≤ 4
Vj(x,t0), j ≤ 4
315
be assumed that there are p coordinate transformations such that the information conveyed
by each of the p characteristic lines concerns only one dependent variable2. In the case
depicted in figure 4.7 p = 2 and the characteristics would be associated to V1 and V3. Thus,
the initial conditions would not be sufficient to specify the solution at P as no information
regarding V2 and V4 would have travelled from earlier times.
Boundary information, i.e., information placed on a time-like line such as x = x0 in figure 4.7
would have to be called to complete the solution. An imprecise generalisation of the notion of
characteristic is often performed. Since the boundary conditions imposed on t ≤ ti, where ti is
the time coordinate of P, affect the solution of P, the horizontal line t = ti is conceived as a
generalised characteristic line. Physically, a horizontal characteristic line means the
information is propagated with infinite velocity. Mathematically, such a horizontal line would
be a space-like line and, as seen in next section, well posed problems do not admit the
specification of boundary conditions on characteristic lines. Thus, the initial condition cannot
specify all the dependent variables.
FIGURE 4.7. Parabolic system with n = 4; domains of influence and of dependence of a given
point P. The solution domain is enclosed by the dashed line .
From the above considerations it is easy to verify that parabolic systems are often associated
to i) systems where not all the time derivatives exist or, in general, where the matrix A in
(4.13) is singular and ii) systems where the initial state is not disturbed by wave-like
phenomena, but whose evolution occurs in time.
The domain of dependence of P is the half-plane { }H ( , ) : i Hx t t t− = ≤ ∩ , as seen in figure
4.7. This is a direct result of the existence of phenomena that propagates with infinite
velocity. The domain of influence of P is H H+ −= , the complement of H. For the problem
idealised in figure 4.7, with n = 4, the initial conditions can specify the only two of the
dependent variables because the x axis is a characteristic line for the remaining two.
Boundary conditions must be introduced, distributed so to match the mathematical and
physical requirements of the system. In general, for the wave-like solutions, the boundary
conditions are placed in time-like lines associated to the b characteristics that satisfy
( ) 0b <C ni (4.20)
2 The circumstances for which this transformation is possible will be discussed below.
x
t
λ(1) ≡ λ(2)
domain of dependence domain of influence
P Η
∂Η
λ(3) ≡ λ(4)
Vj(x,t0) j = 1,3
Η − Η +
316
where n is the outfacing normal to ∂H and C is given by (4.10) (see also Hirsch 1988, p. 99).
For the phenomena that propagate with infinite speed, the boundary information should be
placed in accordance to physical or numeric criteria. In the case shown in figure 4.7, at x = x0
there is one characteristic for which (4.20) holds. At x = x1 it is the characteristic λ(3) that
verifies (4.20). The remaining information was arbitrarily placed at x = x0.
A well-known example of these type of systems can be drawn from river hydraulics. The
shallow water equations without the local inertia in the momentum equation are
( ) ( ) 0t xh uh∂ + ∂ = and ( ) ( )( )212 sin( ) w
x bu gh g h∂ + = β − τ ρ
where the variables assume their usual definition. This is imprecisely called the diffusive
model (Ponce & Simmons 1977) despite the fact that there is no diffusion but propagation
without wave-like character. Only one variable can be specified at t = 0 while the other must
be computed from one of the equations. Usually, the initial condition specifies q(x,0) = u(x,0) h(x,0), the unit discharge. In that case, the water depth, h(x,0), is computed from backwater
equation
( ) ( ){ } ( ) ( )( )2 3 2d 1 ( ,0) sin( ) '( ,0) ( ,0)wx bh q x gh g h q x q x gh− = β − τ ρ −
where ( ) 0'( ,0) dx t
q x q=
= is the derivative of the unit discharge, in case it is not constant. It
is easily seen that the absence of time derivatives in the momentum equation is equivalent to
the assumption of infinite propagation velocities in the channel. In fact, the characteristic
polynomial would yield the roots d 0t = and ( )(1) 2 2d d Fr 1 Frx t u≡ λ = − . The root dt = 0
represents the physical requirement that, at each time level, the flow instantaneously adapts
to any constraint. The finite root depends on the Froude number; the direction of propagation
is positive if Fr > 1 (supercritical regime, upstream hydraulic control) and negative if Fr < 1
(subcritical regime, downstream hydraulic control).
The boundary conditions specify both variables at the upstream reach if ( )2 31 0q gh− < . If
( )2 31 0q gh− > there will be boundary information at both downstream and upstream
reaches. The problem is ill-posed if ( )2 3 1q gh = .
The remaining propagation phenomena are of dispersive or of hyperbolic type (c.f., Whitham
1974, pp. 4-10). In either case, the number of roots of (4.18) is p = m, i.e., there are as many
eigenvalues of 1−A B as dependent variables. Equivalently, the number of independent
propagation directions, or lines of constant phase, is equal to the rank of − λB A .
In the general case the angular frequency is a function of the wave number. If the second
derivatives of the angular frequency are null, as was the case in the derivation of (4.4)
performed in the Annex 4.1, the system is hyperbolic. In this case, the characteristics (phase
velocities) are independent of the wave number since ( ) ( )p pλ = −ϖ κ and
( )pϖ ∝ κ . On the
contrary, in a dispersive system, the second derivatives of the angular frequency are not
zero. It can be shown (Whitham 1974, p. 99) that the individual waves segregate and that the
phase velocity is the propagation velocity of the wave train.
317
Figure 4.8 shows a trivially simple solution domain of a hyperbolic system with n = 4. At each
point P in H, four characteristic directions can be identified. These directions can be tracked
back in time, to the beginning of the times, to form a closed envelop. Such envelope,
represented in light green in figure 4.8, is limited by the “fastest” positive and negative
characteristic lines. It represents the set of points (x,t) whose values of V(x,t) influence the
solution at P = (xi,ti). Because all the propagation speeds are finite, no information that can
affect the solution at P is coming from outside its domain of dependence.
Similarly, there are four characteristic lines issuing from P to future times. The envelope
formed by the fasted positive and negative characteristics, represented in light blue in figure
4.8, is the domain of influence of P. Again as a consequence of the finite propagation
velocities, the solution at P cannot influence any region of H outside this domain.
FIGURE 4.8. Hyperbolic system with n = 4; domains of influence and of dependence of a given
point P. The solution domain is enclosed by the dashed line .
Initial and boundary conditions must be made available in ∂H. The number of initial conditions
is simply the number of intersections between a space-like direction and the characteristic
lines. In fact, this is the basis for another definition of hyperbolicity (c.f., Jeffrey & Taniuty
1964, p. 15). System (4.13), with n dependent variables, is hyperbolic iff any space-like
direction intersects n characteristic lines while satisfying
( ) 0k <C ni , k = 1...n (4.21)
Figure 4.8 shows the intersection of the four characteristic lines with the space-like
boundary. Equation (4.21) is necessary to ensure that the initial conditions are prescribed in
the correct space-like boundary, i.e., the one relative to the past times. A more explicit
depiction of the necessary and sufficient conditions to be prescribed at the proper space-like
boundary is shown in figure 4.9.
The number of boundary conditions in the time-like boundaries is prescribed in accordance to
(4.20). For the simple situation idealised in 4.8 there are three positive characteristics and
one negative. Thus, at a point U in x = x0 there must be 3 independent equations which,
complemented with the information travelling along λ(4), allow for the computation of the
solution V(U). Boundary information at x = x1 is specified in accordance to the same
principles. One equation is required, corresponding to the negative characteristic line. Both
situations are depicted in figure 4.9.
x
t
λ(1) λ(2) λ(3) λ(4)
domain of dependence domain of influence
P Η ∂Η
Η −
Η +
318
FIGURE 4.9. Summary of boundary and initial conditions for a hyperbolic system whose
solution is sought in a simple path-connected set, H, with two time-like and two space-
like boundaries and n = 4.
4.2.1.6 Ill-posedness and characteristics
Consider the PDE ( ) ( )t x∂ + ∂ =V V 0A B . Let V be prescribed at some curve
{ }( , ) : ( ), ( )x t x X t TΒ ≡ ∈ = η = η . Let the implicit functional representation of that curve
be ( )( ), ( ) 0X TΦ η η = .
Then, if both ( )( ), ( )X TΦ = η ηV V and ( )( ), ( )X TΦ η η are known, the tangential derivative
is known to exist if
( )0
( ) dη
Φ ξη = ∂ ξ∫V V
is a Lipschitz continuous bounded function. The derivative is thus defined except at a
countable number of points. Along the curve, one also knows that the directional derivative is
( ) ( ) ( ) ( ) ( )d +dx tX Tη η η∂ = ∂ ∂V V V (4.22)
since the unit vector in the tangential direction is
( ) ( ){ } ( ) ( ) T
d 0lim d d d d d dx tx t X Tη η ηη→
⎡ ⎤= η + η = ⎣ ⎦e e e
In the direction normal to the curve, the derivative of ΦV is unspecified and unknown. Yet,
since the direction normal to eη is, ( ) ( ) Td dT Xν η η⎡ ⎤= −⎣ ⎦e , the directional derivative can
be written
( ) ( ) ( ) ( ) ( )d +dx tT Xν η η∂ = − ∂ ∂V V V (4.23)
It is now searched on which lines does the specification of V allow for the computation of the
normal derivatives and, hence, the partial derivatives w.r.t. x and t. The system of equations
to be solved is, in matricial form
λ(4)
U
3 independent boundary conditions
λ(2) λ(3)
D
λ(1)
∂H
I
H
1 boundary condition
4 independent initial conditions
space-like
319
( ) ( )( ) ( )
( )( )( )
( )d d ddd d
t
xT X
X Tη η η
νη η
⎡ ⎤ ⎡ ⎤∂ =⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥∂⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− − ⎣ ⎦ ⎣ ⎦⎣ ⎦
V 0V VV 0
A B 0I I 0I I I
(4.24)
It is clear that the system has a unique solution iff
( )( ) ( )( )( )det d d 0X Tη η− − − ≠A B (4.25)
If ( )d 0Tη ≠ , which means simply that η ≠ x, then
( )( )( )d
ddet 0XT
η
η− ≠B A (4.26)
Assuming that the function are injective in certain intervals, then, at least locally,
( ) ( )( )
ddd X
t TX η
η=
and
( )( )det d 0t X− ≠B A (4.27)
It was seen that there is one class of curves for which
( )det 0− λ =B A
These are the characteristic curves, whose directions are λ. Thus, in order to make sure that
system (4.24) has a solution, it is necessary that ( )dt X ≠ λ . It is thus concluded that if
( ) ( )( ), ( ) ( ), ( )X T X TΦ η η = Σ η η is a characteristic curve, the specification of V renders the
problem ill-posed, in the sense that (4.24) does not have a unique solution.
As a corollary, boundary or initial conditions cannot be specified over a characteristic line (in
the context of the shallow water/ Exner equations, cf. Ferreira 1998, p. 45).
4.2.1.7 Characteristic variables and compatibility equation
Further attention is now given to the actual computation of the solution of hyperbolic
problems. In a n-dimensional hyperbolic problem, the solution of (4.13) at a point P in the
interior of H can, at least formally, be written as a superposition of n wave forms. It could be
written as
( )
( ) ( )i ( )0
1 1
( , ) ( , ) ( , )ep
p p
n n
x t
p p
x t x t x tκ −λ
= =
= =∑ ∑V V V (4.28)
It was seen that n independent characteristic directions cross at P (figure 4.8) each carrying
independent and complementary information. The recombination of that information provides
the necessary and sufficient conditions to compute the solution at P. Obviously, it is implied
that, in general, each characteristic line carries information concerning all the dependent
variable with physical meaning, i.e., the primitive and the conservative variables.
320
Thus, it is legitimate to ask if there is a transformation of variables (alluded to while
discussing parabolic systems) for which each characteristic line conveys information related
to one variable only.
One looks for solutions in the form of (1) (1) T... 0w= ⎡ ⎤⎣ ⎦W … ( ) ( )0 ...n nw= ⎡ ⎤⎣ ⎦W that
satisfy (4.28). Equivalently, it can be stated that a transformation, not necessarily linear, and
a new set of variables (1) ( ) T... nw w= ⎡ ⎤⎣ ⎦W are sought, such that (4.28) becomes
( )
( )
(1)1(1) (1)
( )( ) ( )
i1 0
i0
( , ) ( , ) ( , )...
( , ) ( , ) ( , )
e
en
nn n
x t
x tn
W x t w x t w x t
W x t w x t w x t
μ −λ
μ −λ
= =
= =
(4.29)
If W exists, along each characteristic line p there travels information regarding Wp ≡ w(p)
only. The quasi-linear system of PDEs would then be amenable to a decomposition such that
its solution would be the superimposition of solutions of scalar quasi-linear equations. To
solve these equations, analogies drawn from the linear scalar equation (see p. 307) are useful.
It will be discussed next how and when the intended decomposition is possible. It will be seen
that while the set of quasi-linear scalar differential equations may be derived for all
hyperbolic systems, not all will allow for explicit determination of W.
In order to find the transformation between V and W, one might use the concept underlying
(4.28) and look for linear combinations of the equations that compose (4.13). Then, a
combination of derivatives is searched such that it is equivalent to the derivative of the
desired new set of variables. The linear combinations of the equations that compose (4.13)
may be written as
( ) ( )( )
( ) ( )
( )
( ) ( )
p
p p
t x
t x
∂ + ∂ = ⇔
⇔ ∂ + ∂ =
l V V 0
a V b V 0
A B
(4.30)
provided that ( ) ( )p p=l aA and
( ) ( )p pl bB = . System (4.30) is a set of p = n PDEs, each
composed of n derivatives of the n dependent variables.
Each equation (4.30) represents also a directional derivative in the space-time domain. The
coefficients a(p) and b(p) may be written in such a way that the direction of the derivative is
made explicit. Without loss of generality, let the direction of the derivatives be taken as the
tangent to the path Γ(x,t) = cte. As usual, let this path be parameterized for s such that
( )x X s= and ( )t T s= . System (4.30) becomes
( ) ( ) ( ) ( )( ) ( )d d 0p ps t s xT X∂ + ∂ =ξ V ξ V (4.31)
whenever
( )( ) ( )dp ps T=l ξA ; ( )( ) ( )dp p
s X=l ξB (4.32)(a)(b)
At this point it is convenient to show that the path Γ(x,t) = cte, whose tangents are the
directions for which the directional derivatives (4.30) are taken, is a characteristic line.
Solving (4.32)(a) and (b) for ( )pξ and equalling the result, it is obtained
321
( ) ( )( )( ) d dps sT X− =l 0B A (4.33)
Assuming that both ( )x X s= and ( )t T s= are injective Lipschitz continuous mappings in
some neighbourhood of (xi,ti), (4.33) becomes, from the implicit function theorem
( )( )( ) dpt X− =l 0B A (4.34)
Equation (4.34) expresses that the vectors l(p) are the left eigenvectors of 1−A B . Non-trivial
solutions for l(p) are possible if the matrix is singular, i.e., if ( )( )det d 0t X− =B A . Two
conclusions can be drawn from (4.34): i) its non-trivial solutions lead to the same eigenvalue
problem, i.e., to the same characteristic polynomial, that was early obtained and expressed in
(4.18); ii) the coefficients, organised in the vector l(p), of the linear combinations of PDEs,
system (4.30), are the entries of the right eigenvectors of 1−A B .
Thus, the only directions that enable rewriting system (4.13) as a combination of derivatives,
all taken in the same direction, are the directions ( ) ( )d pt X = λ of the characteristic lines.
This proposition actually represents another, broader, definition of hyperbolicity. System
(4.13) is totally hyperbolic if the eigenspace of 1−A B is of the same dimension of the space
of the dependent variables, i.e., the system admits as many linearly independent eigenvectors
as dependent variables. According to this definition, a nxn system of PDEs that have less than
n eigenvalues is still hyperbolic if it has n independent eigenvectors (for an example cf.
Whitham 1974, p. 76).
Let (4.31) be written so as to highlight the fact that the derivatives of the dependent variables
are being taken along the characteristic lines. Without loss of generality let t ≡ s as in p. 311.
The following notation for the combination of derivatives can be used
( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( ){ }
( ) ( )
( ) ( )
( ) ( )
1 11
11
d d ... d d 0
D ... D 0
p p
p p
t t t x n t t n t x n
t n t n
T V X V T V X V
V V
ξ ∂ + ∂ + + ξ ∂ + ∂ =
ξ + + ξ =
(4.35)
The derivative Dt(Vk) is the time derivative taken along ( ) ( )d pt X = λ and it can be
interpreted as a Lagrangian derivative. It should be noted that (4.35) is the differential
analogous to (4.28). The meaning of both formulations is that the solution at a point P in H is
achieved through the composition of n independent sources of information, each, in general,
carrying information related to the n dependent variables.
Equation (4.35) is called the compatibility equation of (4.13). The coefficients ( )pkξ are easily
obtained from (4.33) up to an arbitrary scale factor. What is left to know is whether or not
there is a combination of derivatives of V and respective ( )pkξ such that new variables W can
be obtained. For that purpose, the compatibility equation (4.35) can be written as
( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )1 11 1d ... +d ... 0p p p p
t t n t n t x n x nT V V X V Vξ ∂ + + ξ ∂ ξ ∂ + + ξ ∂ = (4.36)
and, then
( ) ( ) ( ) ( )
( ) ( )( )
d +d 0
+ 0p
t t p t x p
t p x p
T W X W
W W
∂ ∂ =
∂ λ ∂ =
(4.37)
322
provided that
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )1 11 1... ; ...p p p p
x n x n x p t n t n t pV V W V V Wξ ∂ + + ξ ∂ = ∂ ξ ∂ + + ξ ∂ = ∂ (4.38)(a)(b)
for each p-characteristic. Equation (4.37) can be further simplified for
( )D =0t pW along ( ) ( )d pt X = λ (4.39)(a)(b)
The meaning of (4.39) is clear: each variable Wp, herein called characteristic variable, does
not change along the corresponding characteristic line. The analogy of the linear scalar
advection equation is now pertinent. Along each family of lines of constant phase
(characteristic lines) there is one and only one variable whose value remains constant along
that direction. Furthermore, the new system is well-posed because the system is hyperbolic
and there are as many variables Wp as characteristic directions.
4.2.1.8 Examples of computation using characteristic variables
Knowing that there are n independent eigenvectors and that the combination of derivatives
taken along characteristic lines may allow for the definition of a new set of variables -
characteristic variables -, let the procedures that led to (4.39) be condensed and rewritten in
a simplified way. Assuming that A is non singular, let M = A−1B. Then, (4.13) becomes
( ) ( )t x∂ + ∂ =V V 0M (4.40)
let ( ) ( )1t t
− ∂ = ∂V WS and ( ) ( )1x x
− ∂ = ∂V WS . Then (4.40) can be written
( ) ( )t x∂ + ∂ =W W 0S MS (4.41)
which leads to
( ) ( )( ) ( )
1 1t x
t x
− −∂ + ∂ =
∂ + ∂ =
W W 0
W W 0
S S S MSΛ
(4.42)
where 1−=Λ S MS . If A is non-singular and x(s) and t(s) are indeed injective Lipschitz
continuous mappings in some neighbourhood of (xi,ti), without loss of generality, it can be
taken ( ) ( )p p=ξ l . In that case the transformation matrix S is, from equation (4.38), defined as
(1)
( )
1 T
T
...n
− ⎡ ⎤=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
l
l
S
(4.43)
From elemental algebra, it is known that there are vectors r such that
( ) ( )
( ) ( )
1 if 0 if
k l
k l
k lk l
= =⎧⎨
= ≠⎩
r lr lii
(4.44)
These are the right eigenvectors of 1−A B , directly obtained from ( )( ) ( )d pt X− =r 0B A .
The transformation matrix can more easily be written as (1) ( )... n= ⎡ ⎤⎣ ⎦r rS .
323
From (4.37) and (4.38) it follows that the transformation 1−=Λ S MS renders necessarily a
diagonal matrix whose main diagonal is composed by the eigenvalues of the system.
Expanding the last equation of (4.42), one verifies that the set of characteristic variables
allows for a formally decoupled system of PDEs, equivalent to (4.13), that reads
( ) ( )
( ) ( )
(1)
( )
1 1 0...
0n
t x
t n x n
W W
W W
∂ + λ ∂ =
∂ + λ ∂ =
(4.45)
As seen above, along the direction ( ) ( )d pt X = λ (4.45) can be written as
( )
( )
1D 0...D 0
t
t n
W
W
=
=
(4.46)
It was showed that the problem of finding a transformation of variables and of coordinates
such that each characteristic line conveys information regarding one variable only may have a
solution. At a point P in H, the solution may be found through (4.46), a set of equations that is
valid along the n characteristic lines that intersect at P. Formally, this is equivalent to what is
expressed in (4.29): the information regarding each characteristic variables is transported by
the respective characteristic line alone. Formally, this is an improvement relatively to what is
expressed in (4.28) or in (4.35). These express that the solution is obtained at the cross of n
lines of constant phase, each of which, in general variables, conveys information related to
the n dependent variables.
Summing up the above discussion, it can be said that the problem of finding a transformation
of variables and of coordinates has a solution if i) the system is totally hyperbolic and ii) if
(4.38) has a solution, not necessarily unique. Assuming that the characteristic lines have the
regularity properties assumed above, (4.38) becomes
( ) ( ) ( )1 1 s s− −∂ = ∂ ⇔ ∂ =VW V WS S (4.47)
Not all nxn equations in (4.47) are independent. Nevertheless, a system of n equations to n
unknowns can be derived. The usual choice is
( ) ( ) ( )
( ) ( ) ( )
1 2
1 2
1 1 11 1 1 11 12 1
1 1 11 2
... ...
...
... ...
n
n
V V V n
V n V n V n n n nn
W W W
W W W
− − −
− − −
∂ + ∂ + + ∂ = + + +
∂ + ∂ + + ∂ = + + +
S S S
S S S
(4.48)
If system (4.48) has a solution, the problem of reducing (4.13) to a quasi-decoupled system of
scalar quasi-linear equations has a simple solution. Unfortunately, there is no guarantee that
(4.48) does have a solution for n > 2. In fact, it is possible to find solutions for (4.48) only for
n ≤ 2 (cf. Hirsch 1988, p. 566). It will be seen later that, in the absence of characteristic
variables, it is always possible to write the compatibility equation (4.35). Its analytical-
numerical solution is also generally well defined, namely at the boundaries. The analytical-
numerical procedures that make use of (4.35) are broadly called method of characteristics. An
324
application of the method of characteristics is shown in Chapter 5, while solving the
governing equations derived in Chapter 2.
The following example reports one case where the solution is easily found. It is the case of
the shallow water equations for the flow of an incompressible fluid over a horizontal, fixed,
perfectly smooth bed.
Example 4.1
Written in the primitive variable non-conservative form, the system of conservation laws
obtained from the depth-integration of the incompressible Navier-Stokes equations, given
appropriate cinematic boundary conditions and in accordance to the shallow-water
approximation, is
( ) ( ) ( ) 0t x xh u h h u∂ + ∂ + ∂ =
( ) ( ) ( ) 0t x xu g h u u∂ + ∂ + ∂ =
The eigenvalues are (1) u ghλ = + and
(2) u ghλ = − . The corresponding left eigenvectors
are (1) 1g h⎡ ⎤= ⎣ ⎦l and
(2) 1g h⎡ ⎤= −⎣ ⎦l . Thus, the inverse and the transformation
matrixes are
1 1
1
g h
g h
− ⎡ ⎤=⎢ ⎥
−⎢ ⎥⎣ ⎦
S
;
12
1 1hg
g h g h
⎡ ⎤=⎢ ⎥
−⎢ ⎥⎣ ⎦
S
Equations (4.48) are, in this case
( ) ( )
( ) ( )
1 1
2 2
1
1
h u
h u
W W h g
W W h g
∂ + ∂ = +
∂ + ∂ = −
(4.49)
Admitting that the derivatives can be separated, W1 can be integrated first in h. It is obtained
( ) ( )(1)1 1 2h W g h W gh f u∂ = ⇔ = +
Deriving the expression for W1, obtained above, with respect to u and integrating the result,
one obtains the first characteristic variable
( ) ( )(1)
(1)
1
1
d 1
2
u uW f
W w u gh
∂ = = ⇔
⇔ ≡ = +
The second characteristic variable can be derived using the same procedures. The result is
(2)
2 2W w u gh≡ = −
It is thus retrieved the well-known result that the shallow water equations can be written as
(1)2u gh K+ = along
(1) u ghλ = +
and
(2)2u gh K− = along
(2) u ghλ = −
325
where K(1) and K(2)
are constants that can be quantified from the initial conditions.
The following example provides further insights on the solution procedure, using the concept
of wave form. The solution procedure for the scalar linear advection equation is used as an
analogy.
Example 4.2
Consider the following system of PDEs, written in conservative form
( ) ( )1 1 2 0t xu u u∂ + ∂ =
( ) ( )( )211 2 1 22 0t xu u u u∂ − − ∂ − = (4.50)(a)(b)
The corresponding characteristic polynomial is ( )( ) ( )( )1 2 1 2 0u u u uλ − + λ + − = from which
the eigenvalues (1)
1 2u uλ = + and ( )(2)1 2u uλ = − − are obtained. The diagonal matrix in
(4.42) is
( )(1)
(2)
1 2
1 2
0000
u uu u
+= = ⎡ ⎤λ⎡ ⎤⎢ ⎥⎢ ⎥ − −λ⎣ ⎦ ⎣ ⎦
Λ
and the right eigenvectors are [ ]T(1) 1 1=r and [ ]T(2) 1 1= −r . The transformation matrix
and its inverse are
1 11 1
= ⎡ ⎤⎢ ⎥−⎣ ⎦
S
;
112 1 1
1 1
− ⎡ ⎤=⎢ ⎥−⎣ ⎦
S
In order to find the characteristic variables, equations (4.47) and (4.48) are invoked. The
characteristic variables are obtained from the following steps
( )( ) ( )
( )
(1)
1
(1) (1)
2 2
(1)
1 11 1 1 22 2
1 11 2 22 2
11 1 22
( )
d ( )u
u u
W W u u
W u u
W w u u
∂ = ⇔ = + ϕ
∂ = ϕ = ⇔ ϕ =
≡ = +
(4.51)
( )( ) ( )
( )
(2)
1
(2) (2)
2 2
(2)
1 12 2 1 22 2
1 12 2 22 2
12 1 22
( )
d ( )u
u u
W W u u
W U U
W w u u
∂ = ⇔ = + ϕ
∂ = ϕ = − ⇔ ϕ = −
≡ = −
(4.52)
System (4.50) can now be written in the characteristic variable formulation. Attending to the
formulation of the characteristics of the system, the following representations are equivalent
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
(1) (1) (1) (1) (1) (1)
(2) (2) (2) (2) (2) (2)
0 2 0
0 2 0
t x t x
t x t x
w w w w w
w w w w w
∂ + λ ∂ = ⇔ ∂ + ∂ =
∂ + λ ∂ = ⇔ ∂ − ∂ =
(4.53)(a)(b)
In order to find the solution of (4.53) one can profit from the fact that the system is truly
decoupled, i.e., each of the equations is a scalar quasi-linear PDE in the characteristic
variable formulation.
326
Consider the following Cauchy problem. The governing equations are (4.50) and the initial
conditions are u1(x,0) and u2(x,0), x ∈ Ω ≡ [0,1]. The latter functions are depicted in figure
4.10. From the data in figure 4.10, the initial values of the characteristic variables are
computed, attending to (4.51) and (4.52).
-1.0
0.0
1.0
2.0
3.0
0 0.5 1x (L)
u1
(-)
-1.0
0.0
1.0
2.0
3.0
0 0.5 1x (L)
u2
(-)
FIGURE 4.10. Initial conditions for the system (4.50) in example 4.2.
The equivalent Cauchy problem, in terms of the characteristic variables, is composed of
(4.53) and the initial data depicted in figures 4.11(a) and (b). A Fourier decomposition of the
initial conditions for w(1) and w(2) is performed, in order to choose an appropriate wave
number for the wave form description. In a one-dimensional domain, the choice of the wave
number cannot influence the space-like direction of propagation, as this is obviously the x
direction. The wave-number is chosen to simplify the expressions of ( )0
pV and of the phase in
(4.14) or of ( )0
pw and respective phase in (4.29).
Figures 4.11(b) and (c) show the Fourier decomposition of the functions shown in figure 4.11.
It is clear that the dominant modes for (1)w are around k = 3 and k = 6. As for
(2)w , the
dominant modes are around k = 3. According to the procedures explained in Annex 4.1, the
wave number for the wave form associated to (1)w is κ = 3. For simplicity, this same value is
chosen for the wave form associated to (2)w .
The solution of (4.53), can be obtained graphically with the help of the scalar linear advection
analogy. The initial conditions are interpreted as wave forms and the value of the phase is
computed in the domain, Ω, of the initial condition. The values of the initial conditions, ( )( ,0)kw x , k
= 1,2, and the respective phases, computed from (4.6), are plotted in figure 4.11. In the same
figure, the associated values of the ( )0
kw , computed from (4.29), are also plotted.
The characteristics, (1)
1 2u uλ = + and ( )(2)1 2u uλ = − − , are computed at each point of Ω. From
each point x ∈ Ω, two lines, corresponding to the two characteristics fields, are issued. Along each
of these lines, the corresponding characteristic variables remain constant, i.e., ( ) ( ) ( )( , ) ( d , d )k k kw x t w x t t t= − λ − , k = 1,2, a result that follows (4.46). Thus, if at t = 0 and x = x0, (1) (1)
0 0( ,0)w x w≡ , then, at t = t +dt, one has (1) (1) (1)1
0 0 02( d ,d )w x w t t w+ = because the characteristics
of the first characteristic field are, in this example, (1) (1)2wλ = . Similarly, for the second
characteristic field, (2) (2)
0 0( ,0)w x w≡ and (2) (2) (2)1
0 0 02( d ,d )w x w t t w− = , because, for the second
characteristic field, and (2) (2)2wλ = − .
327
-2.0
-1.0
0.0
1.0
2.0
0 0.5 1x (L)
w1 (
-)
-2.0
-1.0
0.0
1.0
2.0
0 0.5 1x (L)
w2 (
-)
0
0.5
1
1.5
2
2.5
3
0 10 20 30k (1/L)
w0 (
-)
0
0.5
1
1.5
2
2.5
3
0 10 20 30k (1/L)
w0 (
-)
FIGURE 4.11. Initial conditions for the system (4.53) in example 4.2. Initial values of (a) w(1)
and (b) w(2). Fourier decomposition, in terms of wave numbers and amplitudes, of the
modes that make up the initial conditions of (c) w(1) and (d) w(2).
So far, the solution procedure is analogous to that of the scalar linear advection equation. The
result of this similar procedure is, nonetheless, quite different, as the resulting profiles, after
a given elapsed time, are seen to deform. This feature is clearly seen in figure 4.12. After a
given elapsed time, the profile of the solution of w(1) is sharpened in the positive direction
whereas the profile of w(2) is sharpened in the negative direction. This is a consequence of
the application of the method of constant phase to a system whose characteristics are
proportional to the values of the variables. The constructions on figure 4.12 show that the
value of the phase at a given point x0 will be at x0 + λt and that the larger the value of λ the
farther away that particular value will be found. The value of w associated with that phase
will remain constant along the line of constant phase, thus, higher values of w will travel
faster than lower values of w. Thus, the profiles of ( )kw will sharpen their gradients in the
direction of the respective k-characteristic, as is clear from figure 4.12.
Finally, the solution in terms of the original variables u1 and u2 can be obtained from the
profiles of the characteristic variables. The required transformation is the inverse of that
represented by equations (4.51) and (4.52). In this case, the inversion poses no problems
because it is linear. The results are seen in figure 4.13.
(a) (b)
(d) (c)
328
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.2 0.4 0.6 0.8 1
S (-)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.2 0.4 0.6 0.8 1
S (-)
-1.0
0.0
1.0
2.0
3.0
0 0.2 0.4 0.6 0.8 1
w1
(-)
-1.0
0.0
1.0
2.0
3.0
0 0.2 0.4 0.6 0.8 1
w1
(-)
-1.0
0.0
1.0
2.0
3.0
0 0.2 0.4 0.6 0.8 1x (L)
w10
(-)
-1.0
0.0
1.0
2.0
3.0
0 0.2 0.4 0.6 0.8 1x (L)
w10
(-)
FIGURE 4.12. Construction of the solution of (4.53), example 2. Left: first characteristic
variable; right: second characteristic variable. From top to bottom: (a) phase, (b) wave
form and (c) ( )0
kw . The thin lines ( ) stand for the initial conditions and the thick
lines ( ) stand for the final solutions. Constructions show how the method of
constant phase allows for the construction of the solution.
It is a matter of some interest to observe that there is strong deformation of the wave forms
with apparent attenuation of the wave maximum. Earlier, it was stated that hyperbolicity is
associated to wave propagation without attenuation. This is true for each of the elemental
wave forms that compose (4.28) or (4.29). This can be verified by looking at figure 4.12. Both
wave forms of (1)w and
(2)w are indeed propagating without attenuation. It is the
superposition of the wave forms that causes the change in the maximum amplitudes. This is
easy to understand observing the transformation of the characteristic variables into primitives
ones, equations (4.51) and (4.52), as these operations involve explicit summations and
subtractions.
(a)
(b)
(c)
329
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.2 0.4 0.6 0.8 1
u1 (
-)
t = 0.0 t = 0.12t = 0.24
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 0.2 0.4 0.6 0.8 1x (L)
u2 (
-) t = 0.0t = 0.13
t = 0.24
FIGURE 4.13. Solution of the system (4.50) of example 2. Results at three distinct times for (a)
u1 and (b) u2.
It should also be kept in mind that the functions ( )0
pV represent also phase averages. Thus,
the superposition of non-attenuating elements in equation (4.28) will generally result on
deformation and attenuation.
4.2.1.9 Non-linear wave propagation and shock formation; the scalar case
One last remark concerning non-linear hyperbolic wave propagation should be made at this
stage. It was seen that the waves of example 2 deform and grow steeper in the direction of
its characteristics. This is true because the characteristics are proportional to the value of
the solution at each point of the domain. Later, this condition will be generalised. For now, it
its sufficient to bear in mind that the behaviour shown in example 2 occurs necessarily when
the modulus of the flux function is convex, i.e., ( )2d ( ) 0w f w > . This is the case of both
( ) ( )(1) (1) 2f w w= and ( ) ( )(2) (2) 2f w w= − .
It was seen in figure 4.12(a) that the values of the phase along the x-axis, initially a straight
line, deform as time grows. Eventually, there is point, to the right of the x-axis for w(1) and to
the left in the case of w(2), for which there will be more than one value of the phase, i.e., the
thick lines in figure 4.12(a) will become vertical.
(a)
(b)
330
In fact, because the fluxes in equation (4.53) are convex, the characteristics are increasing
functions of (1)w and
(2)w . In the space time domain, this is represented by the convergence
of the characteristic lines until eventually intercepting at a finite time. This is shown in
figures 4.14(a) and (b), for the case of λ(1)-characteristics of example 2. Figure (a) shows the
formation of the shock whereas figure (b) shows two (1)w profiles before and after the
formation of the shock.
0.00
0.10
0.20
0.30
0.40
0.50
t ( T
)
0.0
0.5
1.0
1.5
2.0
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3x (L)
u1 (
-)
t = 0.10 t = 0.40
FIGURE 4.14. Formation of a shock in non-linear hyperbolic wave propagation. (a)
characteristic and shock paths in the xt domain; (b) profiles of (1)w before and after the
formation of the shock.
In the case of the quasi-linear equations of example 2, the time for which the shock is formed
is easy to compute. The time for which the shock is initiated corresponds to the point, in the
space-time domain, where the first two characteristic lines intercept. Let the path of the
shock be represented by the set of points, { }( , ) : ( )x t x s tΓ ≡ ∈ = . Behind the shock the
value of (1)w is designated by w−
. Similarly w+ is the value of
(1)w immediately after the
shock. With the help of a Taylor expansion around ( )w w− ++ it can be proved that the
direction of the shock, ( )d ( )tS s t≡ , is such that ( ) 212 OS w w+ − − +⎛ ⎞= λ + λ + −⎜ ⎟
⎝ ⎠, where
( )w± ±λ = λ (cf., Dafermos 2000, p. 148).
331
Assuming that the shock is sufficiently weak, the above result states that its velocity lies
between the values of the characteristics on each side of the discontinuity. Since the shock
occurs as a consequence of the convergence of the characteristics, i.e., in a region where the
initial condition is monotone decreasing, ( ) ( )(1)0d ( ,0) d 0x xw x = λ ≤ , then, at the shock
formation, 0 0− +λ > λ and 0 0S− +λ > > λ . Let xA be chosen such that λA > S and xB chosen such
that λB < S. Since the initial condition is monotone decreasing in this region, the set of all xA
is disjoint of the set of all xB and there is xm simultaneously supremum of the set of all xA and
infimum of the set of all xB. Thus, the shock is originated over a characteristic λm issued from
xm.
Let (xI,tI) be the coordinates of a point of interception of Aλ and B Aλ < λ over the shock
path. Then, I A A Ix x t= + λ and I B B Ix x t= + λ , where, at the origin of the time, xA is the
point from which λA was issued and xB is the point from which λB was issued. Simultaneously,
{ }( , ) ( , ) : ( )I Ix t x t x t∈ Γ ≡ ∈ = γ , i.e., (xI,tI) belongs to the path of the shock. The time can
be eliminated to obtain
B AI
B A
x xt
−= −
λ − λ
The critical time for the formation of the shock is the minimum of tI and belongs to the set of
all tm such that
( )0
1 1limdm AA m
m A
mx x
xx x m
t−
−λ −λ −→
−
= − = −λ
,
( )0
1 1limdB mB m
B m
mx x
xx x m
t+
+λ −λ +→
−
= − = −λ
(4.54)
Because the initial distribution of the characteristics is continuous differentiable, the left and
right derivatives given by (4.54) are equal and
( ){ } 10dm x m
t−
= − λ (4.55)
In order to locate xm, the following argument can be pursued. Any point in the region of
monotone decreasing initial conditions is a candidate and must be tested. The shock is
initiated when the characteristics first cross. This means that one is looking for the minimum
tm, computed by (4.55), that is allowed by the initial data. Since, in this region
( ) ( )(1)0d ( ,0) d 0x xw x ∝ λ ≤ , this corresponds to the maximum absolute value of ( )0dx λ or,
equivalently, its necessarily negative minimum. Thus, the time for which a local shock is
formed is
( ){ } 10min dc xt −
= − λ (4.56)
A more general deduction of tc is given in Rhee et al. (1989), p. 45-47.
4.2.1.10 Weak solutions, shocks and simple rarefaction waves
Once formed, the shock will endure permanently in the solution because there is no
mechanism to dampen it away. The solution of (4.13) is called regular while is smooth. Once a
shock is developed, or if the initial conditions are discontinuous, the derivatives in (4.13) are
332
ill-defined. An extended definition of solution is required in order to accommodate its non-
smoothness. A weak solution V(x,t) of the conservation law
( ) ( )( ) ( )t x∂ + ∂ =U V F V 0 (4.57)
is a bounded measurable vector valued function that satisfies
( ) ( ){ } 0H
d d dt x x t x∞
∂ φ + ∂ φ = φ∫ ∫U F V (4.58)
for all smooth test functions φ with compact support, i.e., ( )0C H∞∞φ ∈ , with
{ }H , : 0x t x t∞ ≡ −∞ < < ∞ ∧ ≥ . Because φ is zero outside and at the boundary of the
solution domain, H (see figure 4.8), equation (4.58) becomes
( ) ( ){ }H
t x∂ φ + ∂ φ =∫ U F 0 (4.59)
The solution domain can be divided in two regions separated by the path of the shock. Let
these two disjoint regions be { }LH ( , ) H : ( )x y x s t≡ ∈ < and { }RH ( , ) H : ( )x y x s t≡ ∈ > .
Equation (4.59) can be written
( ) ( ){ } ( ) ( ){ }L RH H
d d d dt x t xx t x t∂ φ + ∂ φ + ∂ φ + ∂ φ =∫ ∫U F U F 0 (4.60)
Each of the above integrals obeys to
( ) ( ){ }
( ) ( ){ } ( ) ( ){ }m
m
H
H
d d
d d
t x
t x t x
x t
x t
∂ φ + ∂ φ =
∂ φ + ∂ φ − ∂ + ∂
∫∫
U F
U F U FmH
d dx tφ∫
(4.61)
The second integral in the right hand side of (4.61) is zero because its integrand is (4.57), the
differential conservation law, and V is at least C1 in Hm, m ≡ L or m ≡ R. The first integral on
the right hand side can be expanded with the help of Green’s theorem3 in the plane. For
instance, for the region left of the path of the shock, one has
( ) ( ){ } ( ) ( ) ( ){ }L L L
L L
H H H
H H
d d d dt x x t x t∂ ∂ ∂
∂
∂ φ + ∂ φ = φ −∫ ∫U F U F (4.62)
3 For the purposes of this text, Green’s theorem for two variables states that (Apostol 1980, p. 380)
( ) ( ){ } { }1 2 1 2 1 2
H H
d d d dx xu v x x u x v x∂
∂ − ∂ = +∫ ∫
333
The circulation integral in (4.62) is evaluated considering that φ is necessarily zero in the
“exterior” boundary of HL but might be different from zero in the boundary between HL and
HR. Thus
( ) ( ) ( ){ }
{ }
L L L
L
H H H
H
0
d d
( ( ), )d ( ( ), )d
x t
s t t x s t t t
∂ ∂ ∂
∂
+ + +
Γ↑
φ − =
φ − + φ
∫∫
U F
U F ( ) ( ){ }L L
L
H H
H \
d dx t∂ ∂
∂ Γ
−∫ U F
and
( ) ( ){ } { }LH
d d ( ( ), )d ( ( ), )dt x x t s t t x s t t t+ + +
Γ↑
∂ φ + ∂ φ = φ −∫ ∫U F U F (4.63)
A change of variables yields
{ } ( ){ }( ( ), )d ( ( ), )d ( ( ), )d ( ( ), ) dtt
s t t x s t t t s t t s s t t t+ + + + + +
Γ↑ Δ
φ − = φ −∫ ∫U F U F (4.64)
Likewise, in the region to the right of the path of the shock one has
{ } ( ){ }( ( ), )d ( ( ), )d ( ( ), )d ( ( ), ) dtt
s t t x s t t t s t t s s t t t− − − − − −
Γ↓ Δ
φ − = − φ −∫ ∫U F U F (4.65)
since φ is continuous across the shock path (it should not be forgotten that ( )0C H∞φ ∈ ) then
φ+ = φ− . Equalling (4.64) and (4.65) one obtains
( ) ( )S+ − + −− − − =U U F F 0 (4.66)
where ( )( ),s t t± ±=U U , ( )( ),s t t± ±=F F and the velocity of the shock is, as seen before,
( )dtS s= . The system of equations (4.66) is known as the Rankine-Hugoniot shock
conditions. It governs the solution of (4.57) in the points in which its derivatives do not make
sense.
In the case of the quasi-linear scalar equation the shock velocity is obtained from
( ) ( ) ( )( )( )
( )f w
S w f w Sw
ΔΔ = Δ ⇔ =
Δ (4.67)
For the case of the particular equation whose solution is depicted in figure 4.14, equation
(4.53)(a), the velocity of the shock is
( )
( ) ( )2 2
L RL R 1L R L R2
L R L R
w wf fS w w
w w w w
−−= = = + = λ + λ
− − (4.68)
This result confirms the assumptions made in the derivation of the time corresponding to the
initiating of the shock.
334
One last remark concerns the unicity of the solution. Once discontinuities are developed,
weak solutions are potentially non-unique. The following classical result (cf. Prasad 2001, p.
12) highlights one such case. Consider equation ( ) ( ) 0t xw w w∂ + ∂ = , with initial conditions
1, 0
( ,0)1, > 0
xw x
x− ≤⎧
= ⎨⎩
(4.69)
An admissible solution is
1,
( , ) ,
1,
xt
x t
w x t t x t
x t
− ≤ −⎧⎪
= − < ≤⎨⎪ >⎩
(4.70)(a)
as it verifies the differential equation for all times larger than zero. But it is also easily
verifiable that
1,
,
, 0( , )
, 0
,
1,
xt
xt
x t
t x at
a at xw x t
a x at
at x t
x t
− ≤ −⎧⎪
− < ≤ −⎪⎪− − < ≤⎪= ⎨ < ≤⎪⎪ < ≤⎪⎪ >⎩
(4.70)(b)
with 0 ≤ a ≤ 1. The graphic depiction of (4.70) is shown in figure 4.15.
Solution (4.70)(b) also verifies the governing differential equation in the continuous regions.
Across the stationary shock at x = 0, the solution verifies the Rankine-Hugoniot conditions
(4.68). The solution is thus not unique.
Uniqueness can, in some circumstances, be attained by limiting the set of solutions to those
physically admissible. The work of Olga Oleinik (cf. Magenes 1996) is among the first to
provide a sound basis for the choice of physically admissible solutions. In the scalar case, the
solution of the equation ( ) ( )( ) 0t xw f w∂ + ∂ = is physically admissible if
( ) ( ) ( ) ( ) ( ) ( )f f w f w f w f w f
w w w w
− − + +
− − + +ξ − − − ξ
≥ ≥ξ − − − ξ
(4.71)
for every ξ between w+ and w−
. If it is convex, which is the case for most applications in
this text, (4.71) is equivalent to
S− +λ ≥ ≥ λ (4.72)
This is easily shown by introducing (4.67) in (4.71) and taking the limits w−ξ → and
w+ξ → . The role of ξ in (4.71) is to check the regularity of f(w) between ( )f w+ and
( )f w−. The convexity of f(w) is required in (4.72) to ensure the regularity of the flux. For
general fluxes, it can be imagined that the small variations in the strength of the shock could
render (4.72) false. This is shown in figure 4.16.
335
-1.5-1
-0.50
0.51
1.5
-5 0 5x /t
wa = 0
-1.5-1
-0.50
0.51
1.5
-5 0 5x /t
w
a = 0.5
-1.5-1
-0.50
0.51
1.5
-5 0 5x /t
w
a = 1
FIGURE 4.15. Family of solutions of the Burgers equation for the initial data (4.69)
parameterised by a.
If, for instance, due to a change in the initial conditions, the right state changed from 1w+ to
2w+, then λ− < λ2
+ and (4.72) could become not valid. This would question the well-posedness
of the solution given small variations of the initial conditions.
FIGURE 4.16. Implication of fluxes of ( ) ( )( ) 0t xw f w∂ + ∂ = on the admissibility condition
(4.72). (a) convex flux and (b) non-convex, non-concave flux.
The inequalities (4.72) are known as Lax E-condition. It loosely states that expansive shocks
are not admissible, only compressive ones. The first are depicted in figure 4.17(b) as a set of
rays issued from the shock path. The second are shown in 4.17(a) and were already identified
in figure 4.14; the characteristics converge into the shock path.
A suggestive interpretation recurs again to the information transfer metaphor. In compressive
shocks, the information carried by the incoming characteristics is lost. A different way to
transfer information across the shock is required; this is represented by the Rankine-
Hugoniot conditions (4.66). In the case of expansive shocks, new information is continuously
generated at the shock as if the initial conditions were folded into a line, the shock path, in a
given singular point. It is shown that this production of information would correspond to a
decrease of the entropy of the system, hence its inadmissibility. The details are outside the
scope of this text and can be consulted in, e.g., Dafermos 2000, p. 160-163. It should be
noticed that, in most of the references, the (mathematical) entropy considered is the negative
of the physical entropy. Thus, admissible shocks are, in those references, those for which the
(mathematical) entropy decreases.
w+ w− w
f(w)
f(w+)
f(w−) dw(f(w−)) ≡λ−
dw(f(w+)) ≡λ+ w1
+ w− w
f(w)
f(w1+)
f(w−)λ−
λ1+
λ− > λ+, ∀w+<w−
w2+
λ2+
λ− > λ1+
λ− < λ2+
f(w2+)
(a) (b)
336
FIGURE 4.17. Admissible and non-admissible shocks according to the Lax entropy condition
(4.72). (a) admissible (compressible) shock; (b) non-admissible (expansive) shock.
Clearly, the only admissible solution corresponding to the initial conditions (4.69) is (4.70)(a).
The shock appearing in (4.70)(b) is clearly not compressive as seen in figure 4.18(b). The
blue lines are the characteristics that are issued from the points along the shock path. They
correspond to the constant state around the shock observed in figure 4.15, a = 0.5.
0
0.25
0.5
0.75
1
-2 0 2x /t
t ( T
)
0
0.25
0.5
0.75
1
-2 0 2x /t
t ( T
)
a = 0.5
FIGURE 4.18. Characteristics and shock paths corresponding to (a) solution (4.70)(a) and (b)
solution (4.70)(b), a = 0.5. Blue lines correspond to the characteristics issued from the
shock.
The admissible solution (4.70) could not present any shock, a result that follows from the fact
that w w− +< in the initial condition. No shock would be able to verify (4.72) in the vicinity of
the initial conditions. The admissible solution, (4.70), is, in terms of the characteristic lines,
represented by the fan seen in figure 4.18(a). This configuration is designated an expansive
rarefaction wave. It should be noticed that the solution shown in figure 4.15, a = 0.5, features
a rarefaction wave split by a constant shock at the place where a vertical (zero) characteristic
should be. It is frequent to find numerical discretisations featuring this erroneous solution
when facing the problem of computing a rarefaction wave through a critical flow point (cf.
Ferreira & Leal 1998). In this case, entropy corrections are enforced to keep physically
relevant solutions only (see Hirsch 1988, p. 434 and Chapter 5, §5.3.1 and §5.6.5.5).
The properties of the weak solutions will be discussed at length in §4.3.3 of the present
chapter. For further details on the properties of discontinuous solutions see, e.g., LeVeque
(1990), p. 8, Prasad (2001), p. 23, Toro 1998, p. 64, Dafermos (2000) §8, pp. 147-175 or
LeFloch (1988).
λ−
x
t
x
t λ− > S > λ+
λ+λ−
λ+
λ− > S > λ+ (a) (b)
(a) (b)
337
4.2.1.11 Summary of the notes on hyperbolicity
Hyperbolicity and some properties of non-linear hyperbolic wave propagation were discussed
in the previous paragraphs. The main results will be frequently used in the remainder of this
text. Thus, it is appropriate to summarize the main topics addressed. These can be organized
as follows:
i) hyperbolic propagation represents propagation without attenuation; for a scalar problem the
general solution is ( )( )0( , ) d ( )ww x t w x f w t= + ;
ii) for systems of PDEs, it is noted that the information supplied at the beginning of the times
is conveyed along lines of constant phase (equation (4.6)); for this purpose, it is admitted that
the wave forms are describable by a wave-like description (equation (4.4));
iii) at a given point in the solution domain, the solution is attained by superimposing necessary
and sufficient information; such information travels along the lines of constant phase;
iv) the computation of the direction of the lines of constant phase is amenable to a eigenvalue
problem (expressed in equation (4.18)); the roots of the characteristic polynomial
corresponding to a given system (characteristics of the system) are the directions of the lines
of constant phase;
v) strict hyperbolicity requires that the number of characteristics must be equal to the number
of dependent variables of the system;
vi) a weaker condition for hyperbolicity requires that there are as many independent
eigenvectors as dependent variables;
vii) in some simple 2x2 systems, there are variables, called characteristic variables, such that
its values are propagated along the respective characteristics; for all other variables, each of
the characteristics convey information regarding all the dependent variables;
viii) if a characteristic variable formulation is not feasible, the compatibility equation (4.35)
can always be written; this is at the root of the method of characteristics;
ix) imposing information on a characteristic line renders the problem ill-posed;
x) the initial conditions must specify all dependent variables;
xi) at the boundaries, there are as many equations as negative characteristics relatively to the
exterior normal at that boundary.
xii) compressive shocks will occur if the initial conditions are monotone decreasing and the
flux is convex;
xiii) expansive shocks are not admissible as they represent non-physical sources of
information;
xiv) across a shock, the derivatives of (4.13) or (4.57) are not defined; the solution is
specified by the Rankine-Hugoniot conditions (4.66)
Having addressed the general features of non-linear hyperbolic wave propagation, the
following sections are dedicated to presentation of a summary of the conservation equations,
whose particular properties are to be studied with greater depth with the techniques so far
discussed.
338
4.2.2 Governing equations
4.2.2.1 Equations of conservation
The mathematical model applicable to geomorphic dam-break flows, was developed in
Chapter 3, namely in §3.5, §3.6 and §3.7. It features unsteady flow hydro- and sediment
dynamics and channel morphology and it resorts to semi-empirical formulations to account
for flow resistance and depth-averaged velocity. In what concerns sediment dynamics, the
dense limit approximation of granular flow theory of Jenkins & Richman (1985) and (1988),
rooted in Enskog’s kinetic theory of dense gases (Chapman & Cowling 1970, §16, pp. 297-
322), provided the theoretical background for the core of the model.
Two-dimensional governing equations are integrated in an idealised layered domain shown in
figure 3.32, p. 266. The lowermost layer is the bed, composed of grains with no appreciable
vertical or horizontal mean motion. While the total flow depth and the depth-averaged
velocity are, respectively, h and u, the flow is subdivide into two regions: i) the contact load
layer, whose thickness, depth-averaged velocity and concentration are hc, uc and Cc,
respectively, and ii) the suspended sediment layer, whose thickness, velocity and
concentration are hs = h – hc and us = (uh – uchc)/hs respectively.
The bed is defined as the surface that connects the centres of gravity of the uppermost layer
of the pack of immobile grains. Above, the contact load layer is characterized by high
concentrations and the stresses are generated during collision events among particles. Linear
and angular momentum are transferred mostly by inelastic collisions. The submerged weight
of the sediment grains is equilibrated by a reaction force in the bed, solicited by the quasi-
permanent contacts in the boundary between the transport layer and the bed.
Above the contact load layer, the uppermost flow layer transports a comparatively small
amount of wash load or no sediment at all. Hydrodynamic lift and drag are expected to play
some role in this more diluted region. The grains are not sustained by collisional interactions
but by lift forces originated from turbulent fluctuations (Sumer & Deigaard 1981, Nezu &
Nakagawa 1993, §12, p. 251-258). Turbulent stresses are expected to be dominant in this
region.
This layered structure is expected to retain the essential mechanisms of the two dimensional
conceptual model while being mathematically simpler. In the process of integration of the
governing two-dimensional equations, the shallow flow hypotheses are employed. It is thus
assumed that: i) flow depth is small in comparison to the representative longitudinal length
scale of the flow, ii) local and convective accelerations normal to the flow direction are
negligibly small and iii) slope in the longitudinal direction is small. Conservation equations are
thus found for each of the solid and fluid constituents in each of the identified layers. A
reasonable compromise between computational simplicity and phenomenological complexity
can be achieved if the equations are combined into the following system of five equations to
five unknowns, equations (3.221) to (3.225), shown in p. 278.
Equations (3.221) to (3.225) can be solved for the total flow depth, h = hs + hc, for the layer-
averaged flow velocity ( ) ( )s s c c s cu u h u h h h= + + , for the flux-averaged concentrations Cs and Cc and for the bed elevation Yb. Because of relevance is immediately perceived, the latter
variables form the set of the so-called primitive variables. Closure equations are required for
339
the layer thickness hc and average velocity uc in the contact load layer, for the bed shear
stress, τcb, and for the mass density fluxes between the bed and the contact load layer and
the contact load layer and the suspended sediment layer.
The remaining variables are the average flow density ( )m s s s c c cu h u h uhρ = ρ + ρ or, simply
( )( ) 1 ( 1)wm s Cρ = ρ + − , and the average concentration ( ) ( )s s s c c c s s c cC C u h C u h u h u h= + + .
No special formulations are advanced to account for the saturated flow that may occur in the
wave front, as seen in figures 3.1(a) and (b). However, it is suggested that an earth pressure
coefficient k’c > 1 should be used when the hc = h.
Because of the source terms, equations (3.221) to (3.225) are not amenable to theoretical
treatment. Fraccarollo & Capart (2002) went beyond simply discarding the source terms.
They attempted to show that there is a time window where homogeneous equations drawn
from equations similar to (3.221) to (3.225) describe the phenomena with sufficient accuracy.
They proposed characteristic time scales for the geomorphic and frictional phenomena that
occur in erosional dam-break flows. Before a given characteristic time, the transport capacity
is different from the actual sediment load. That time defines the geomorphic time scale, tg.
Beyond this moment, it is expected that local equilibrium is a valid hypothesis. The frictional
time scale, tf, is the characteristic time beyond which frictional effects become dominant in
the momentum conservation equation.
Under the assumed hypotheses and for the specific set of formulations used, Fraccarollo &
Capart (2002) showed that there is indeed a time window such that g ft t t< < . Furthermore,
they also retrieved theoretically what has been observed in laboratory experiments (cf. Leal
et al. 2003), that, for heavy granular materials like sand, local equilibrium conditions appear
to be always valid, whereas, for light material, there seems to occur non-equilibrium
sediment transport. This is explained by the fact that the characteristic hydrodynamic time
scale, 0 0t h g= , is of the order of magnitude of gt for heavy material, while, for light
material, 0 gt t . In the first case, the shallow-water and the equilibrium hypotheses become
valid at the same time. In the second case, the solution is valid, i.e., the shallow-water
assumptions are valid, before the local equilibrium hypothesis becomes admissible.
Although Fraccarollo & Capart’s (2002) results are strictly valid for erosional flows and for a
particular conception of the frictional time scale, it will be assumed that there is indeed a time
window for which the homogeneous equations are a sufficiently good account of the
geomorphic dam-break flow. Furthermore, given the fact that the concentrations of
suspended sediment are expected to be one order of magnitude smaller than the
concentration of the contact load (cf. Bagnold 1966), Cs will be discarded. Under these
assumptions, system (3.221) to (3.225) can be simplified and written in quasi-conservative
form
( ) ( ) 0t xY uh∂ + ∂ = (4.73)
( ) ( ) ( )( )( ) ( ) ( )2 2 2 212 2w w w
t x c c c s s x s s c c cR u h u h g h h h h∂ + ∂ ρ + ρ + ∂ ρ + ρ + ρ =
( ) ( )( )wc c s x b bcg h h Y− ρ + ρ ∂ − τ (4.74)
( ) ( ) 0t xZ Cuh∂ + ∂ = (4.75)
340
where bY h Y= + is the water elevation, uhR mρ= is the mass discharge per unit width and
(1 ) b c cZ p Y C h= − + is an equivalent bed elevation that takes into account the sediment
stored in the contact load layer. Equations (4.73) to (4.75) are solved for h, u and Yb. No
closures are required for the fluxes normal to the flow but the concentration in the contact
load layer, Cc, must be prescribed. The earth pressure coefficient k’c will not be considered
in this analysis since it is a constant that would not change the nature of the pressure terms.
For the application envisaged in Chapter 2, it was shown, in §2.2.2.6, the momentum
associated to the near-bed sediment transport is negligible if compared to the momentum of
the water flow. In that case, the above equations can be simplified to
( ) ( )( ) ( ) 0t t b x xh Y u h h u∂ + ∂ + ∂ + ∂ = (4.76)
( ) ( )( )( ) 1 (1 ) ( )t s t buu A p C A C C Yh
∂ − + − − + − ∂
( ) ( )( )2
2 12( ) ( ) ( )h s h s x
uA C u gh A C h C C g hh
⎛ ⎞+ ∂ + − ∂ + − + ∂⎜ ⎟⎜ ⎟
⎝ ⎠
( ) ( )( )( )2 12( ) ( ) ( ) ( )u s u s x x bA C u gh uA C u C C u u g Y+ ∂ + − ∂ + − + ∂ + ∂ =
( )( ) (1 ( 1) )wb ss C h− τ ρ + − (4.77)
( )
( ) ( )(1 ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 0t b s h s t u s t
h x u x
p Y C h C h h C u
C C h u h C C u h u
− ∂ + + ∂ ∂ + ∂ ∂
+ + ∂ ∂ + + ∂ ∂ =
(4.78)
in which ( ) ( )1 1 ( 1) sA s s C= − + − allows for the inertia of the suspended sediment to be
considered in the momentum equation. This system admits the unknowns h, u and Yb. Closure
equations are required for Cs, C and τb. The correct use of this model requires that the time
scales of the morphological processes associated to sediment transport are sufficiently low to
admit that the quasi-equilibrium model is sufficiently good.
4.2.2.2 Closure and constitutive equations
Equations for hc, uc and τbc were derived in §3.7.2, pp. 278-287, from the granular flow
theoretical framework that produced the conservation equations. The contact load layer is
divided into a frictional sub-layer, a region were stresses are mainly rate-independent, and a
collisional sub-layer, were stresses are born from momentum exchange due to inter-particle
collisions (figure 3.32). Considering that the dam-break flow is a shear flow in a gravitic field,
it is argued that the normal stresses increase with increasing depth, thus increasing the
sediment concentration and restricting the fluctuating movement. As a result, enduring
contacts among particles become frequent and frictional stresses become predominant in the
lower areas of the contact load layer. Thus, the frictional sub-layer corresponds to the
bottom region of the idealised contact load layer. Two mechanisms govern the dynamics of
the frictional sub-layer: i) increasing the shear rate results in the increase of the sediment
load and, thus, the contact load thickness would increase and, ultimately, so would the normal
stresses at the bottom; as a result, the thickness of frictional sub-layer would increase; ii)
increasing shear rate would also increase the flux of granular temperature towards the
bottom; the kinetic energy per unit time supplied to the fluctuating motion would tend to repel
341
particles and reduce the sediment concentration and the normal stresses; enduring frictional
bonds would be broken and the thickness of the frictional sub-layer would diminish.
Both mechanisms act on the triad: concentration, contact load thickness and granular
temperature. If the equation of conservation of the fluctuating energy is invoked, the granular
temperature can be expressed as a function of the former. Thus, it should be possible to draw
a relation between concentration and contact load thickness from the dynamics of the
frictional sub-layer. A control volume analysis of the momentum exchange in the frictional
sub-layer provided, in §3.7.2, the means to derive the relation between hc and Cc. Assuming
that: i) the superposition of the above described mechanisms render the local inertia
negligible (not the convective inertia), ii) the frictional sub-layer acts as a buffer,
transforming collisional stresses above into frictional ones below and iii) equilibrium occurs
when frictional and collisional stresses cancel on each side of the sub-layer, the searched
relation can be expressed as
tan( )
sc
b c
dC
hθ
=ϕ
(4.79)
where ( ) ( 1)w
bc sg s dθ = τ ρ − is the Shields parameter. The thickness of the contact load
layer was determined from the equation of conservation of the fluctuating kinetic energy. The
resulting expression was compared with the experimental data of Sumer et al. (1996). It was
found that thickness of the contact load layer could be well approximated by
1.7 5.5c
s
hd
= + θ (4.80)
As for the velocity in the contact load layer, it was found that the velocity profile in the
contact load layer would be well approximated by a power law whose leading term would be
( )34
sy d . Its depth integration renders
341
( ) 4107
0
1 ( )d ( 1)c
gh
cc x s
c s
hu u g s d
h d− ⎛ ⎞
= ξ ξ = − θ ⎜ ⎟⎝ ⎠∫ (4.81)
where ( ) ( , )gxu y t is the longitudinal velocity at a height y above the bed.
From dimensional arguments, namely that the shear stress scales with the square of the shear
rate in the collisional granular flow regime (cf. Lun et al. 1984, Jenkins & Richman 1988 or
Campbell 1989), and re-plotting the data of Sumer et al. (1996), the shear stress may be
written as
( ) 2w
bc fC uτ = ρ (4.82)
where the resistance coefficient is given by ( )( )0f f s sC C d h u w= , in an initial stage and,
in a latter stage, ( )( )( )20 1f f s sC u C d h u w= − ϒ + . The involved coefficients must be
determined empirically.
Physical restrictions must be added to equations (4.79) to (4.81) in order to specify Cc, hc and
uc. The depth-averaged sediment concentration cannot exceed 1 – p, where p is the bed
porosity. The thickness and the depth-averaged velocity of the contact load layer are limited
342
by the total flow depth, h, and by the depth-averaged flow velocity, u, respectively. Figure
4.19 shows a virtual sequence of uniform flows, performed with the plastic granular material
of the experiments of Sumer et al. (1996), illustrating the need for restrictions. At a given
flume inclination, the sheet flow occupies most of the flow depth and velocity of the contact
load layer becomes the depth-averaged velocity. From this point on, the flow is saturated and
may be treated as fully mature debris flow. It should be noticed that since the expressions for
hc and uc were developed independently, the saturation point does not occur at the same
Shields parameter in both sequences. This represents but a minor physical imprecision if it is
guaranteed that cu u→ faster than ch h→ , in order to ensure that u is continuous. In this
context, saturation means that the whole of the flow depth is laden with sediment. Thus, when
0cu u +− , one must have 1cC p≤ − .
FIGURE 4.19. Left axis: non-dimensional flow velocity depth-averaged over the contact layer
(uc, ) and over the totality of the flow depth (u, ); velocities are made non-
dimensional by the friction velocity u*. Right axis: non-dimensional thickness of the
contact load layer (hc, ) and total flow depth (h, ); depths made non-dimensional by
the representative grain diameter. Each value of the Shields parameter corresponds to a
virtual uniform sheet flow achieved in a 0.3 m wide with ds = 0.003 m, ws = 0.119 m/s,
s = 1.27, and Cf0 = 0.05. The sequence represents the gradual tilting of the flume from
ic = 0.01 to 0.107.
The main shortcoming posed by the closure equations (4.79) to (4.81) is that they are
expressed by continuous but non-differentiable functions. This implies that the functions that
express for the characteristic speeds are not ( )1+C × , which, as seen later, represents a
serious mathematical problem. Thus, throughout the remainder of the text, for computational
purposes, it will be imposed that cu u= , ( ) 0h cu∂ = , ( ) 1u cu∂ = , ch h= , ( ) 1h ch∂ = and
( ) 0u ch∂ = .
The errors committed are ( )2Ordh hE = ε and ( )2Ordu uE = ε where ( )h ch h hε = − and
( )u cu u uε = − . After the solution is found, the thickness of the transport layer can be
retrieved by (4.80).
The closure equations corresponding to the model based on equations (4.76) to (4.78) are the
following. The thickness of the bedload layer is
1
10
1 10Shields parameter
u/u*
uc
/u*
10
100
h/ds
hc
/ds
h c
h
u
u c
343
{ }* *min ( , , , ), 4b s cr sh d g u u s d= Ψ υ − (4.83)
The velocity and the flux-averaged concentration of sediment in the bedload layer are,
respectively,
( )( )*1 1b wb cbu u C= − α − (4.84)
and
( )2
8 *2
*3.1 10 ( 1)cb s c
h uC s gd−= − θ θ − θ
υ αx (4.85)
where *7.1wbu u= is the velocity of the water in that layer and * wb cbu uα = where cbu is
the velocity of the particles travelling as bedload.
The resistance law is equation (4.82) but the friction coefficient is
( )( )21
2.5ln 5.22f
s
Ch d
=+
(4.86)
This closure sub-model does not require additional physical restrictions because the
thickness of the bedload layer is always smaller than the flow depth. Some care in the
implementation of the equations is required since, in incipient transport conditions, 0cbu ≈
and * 1α . However, in the same conditions, 1cbC ; hence, the product *cbC α in (4.84)
is indeterminate. This problem is tackled by computing the flux-averaged concentration in the
first place and only then the average velocity. If the concentration, given by (4.85), is below
an arbitrarily small number, the velocity is set *7.1b wbu u u= = .
Since both systems of equations, (4.73) to (4.75) and (4.76) to (4.78), express a layered
physical system, its structure is fundamentally the same. The following analysis will be based
on system composed by equations (4.73) to (4.75) because it is this system that is subjected
to the occurrence of discontinuities and, hence, requires conservative formulations.
4.2.3 Conservative and quasi-linear formulations
Written in vector notation, the first order, non-homogeneous, system of PDEs represented by
equations (4.73), (4.74) and (4.75) becomes
( )( ) ( )( ) ( )t x∂ + ∂ =U V F V G V (4.87)
where ] [ 3: 0,× +∞ →V is the vector of dependent primitive variables, 3 3: →U is
defined by [ ]TY R Z=U , where the entries are defined in §2.1, equations (4.73) to (4.75),
3 3: →F is defined as
( )( ) ( ) ( )2 2 2 212 2w w w
c c c s s s s c c c
uh
u h u h g h h h h
Cuh
⎡ ⎤=⎢ ⎥ρ + ρ + ρ + ρ + ρ⎢ ⎥
⎢ ⎥⎢ ⎥⎣ ⎦
F (4.88)
and 3 3: →G is the vector of source terms, defined as
344
( )( ) ( )
0, ;
0b x bf u h Y
= ⎡ ⎤⎢ ⎥− ⋅ ∂⎢ ⎥⎢ ⎥⎣ ⎦
G V
(4.89)
where ( ) ( )( ), ; wb c c sf u h g h h⋅ = ρ + ρ .
Variable t is meant to stand for time, thus physically featuring non-negativity and
irreversibility. It does not necessarily follows that t is mathematically a time-like variable.
For that purpose, it is necessary and sufficient that there is a smooth transformation
:φ V U and a linear operator, δ , such that ( ) ( )δ = δU VA . The latter condition is true
iff 3 3×∈A is non-singular (details at Hirsch 1988, p. 148, although the author requires that
the matrix is positive definite). It follows that x = constant is a time-like surface and that U
and F are of different nature.
To highlight this claim, it is noted that, if G = 0, and if the solution V vanishes outside a
closed bounded interval of , one has
( , )dt−∞
−∞
ξ ξ =∫ U M (4.90)
where M is independent of t, while, on the contrary, it is generally not true that
0
( , )dx+∞
ξ ξ∫ F is independent of x. Physically, (4.90) means that quantities U are conserved
in time and, hence, called the vector of conservative variables, and A is the Jacobian matrix
of the transformation between primitive and conservative variables. Obviously, equation
(4.90) is verified if ] [ 3: 0,× +∞ →V is zero outside some arbitrarily large interval
] [0,Ω ⊂ × +∞ .
In order to show that A is indeed non-singular, let its coefficients be computed accordingly to
the definition ( )= ∂V UA . One obtains
1 2
1 2
1 0 10
1K KN N p
= ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦
A (4.91)
where
( )1 m h mK u hu= ρ + ∂ ρ
( )2 m u mK h hu= ρ + ∂ ρ
( ) ( ) ( ) ( )1 c h c s h s c h c s h sN h C h C C h C h= ∂ + ∂ + ∂ + ∂
( ) ( ) ( ) ( )2 c u c s u s c u c s u sN h C h C C h C h= ∂ + ∂ + ∂ + ∂
Thus, it is clear that A is singular only for the trivial case h = u = 0.
345
The concept of hyperbolicity, in the context of the classification of PDEs, is associated to the
shape of the domains of dependence, P−Ω , and of influence, P
+Ω , of a given point
] [P 0,∈ × +∞ . In a hyperbolic problem, the region of influence of P is delimited by lines that
are neither space-like nor time-like. It should be recalled that given any time-like line, Τ,
there is at least one point P such that P PT − +⊃ Ω Ω∪ . Hence, information travels from the
boundaries (time-like or space-like) is with finite velocity. Also, any problem posed in a
bounded domain for a hyperbolic system is amenable to an initial value (or Cauchy) problem.
A system of conservation laws ( ) ( )t x∂ + ∂ =U F 0 is hyperbolic iff the system of PDEs
( ) ( )t x∂ + ∂ =V V 0A J is hyperbolic, where ( )= ∂V UA is non-singular and ( )= ∂V FJ . As
for the non-homogeneous system, it is necessary to make explicit some of its properties.
While the grouping of the variables in equations (4.73) and (4.75) has no special physical
meaning, equation (4.74) bears visible traces of the control volume analysis from which it was
derived. The conserved variable is R, the unit mass discharge, i.e., the depth-averaged
momentum per unit width and unit channel length. The flux of momentum is dS
u Sρ ⋅∫∫ u n
from which 2
mu hρ is obtained after time and space integration. The material derivative of
the momentum is thus represented by ( ) ( )( )2 2wt x c c c s sR u h u h∂ + ∂ ρ + ρ and, from Newton’s
first law, it is equal to the sum of external forces. These are divided into surface and mass
forces. The former are only represented by the pressure-related terms,
( )( )( ) ( )2 212 2w w
x s s c c cg h h h h∂ ρ + ρ + ρ , since the dissipative forces per unit flow area, τbc, were
discarded. The latter is the force of gravity per unit horizontal area
( ) ( )( )wc c s x bg h h Y− ρ + ρ ∂ . It is assumed that ( )x b y xY g g∂ ≈ throughout the flow depth.
From the above discussion, it is clear that the vector F, designated flux vector, incorporates
the flux of the conservative variables and all the terms related to the external forces that can
be written as a gradient. Similarly, the source term incorporates all the forces that cannot be
written as a simple derivative. Therefore, there is no physical argument underlying the
grouping represented by (4.87), but merely a formal one. On the contrary, there is an
important argument to keep ( )b x bf Y∂ in the system: the fact that it represents a strong
coupling mechanism between geomorphic and hydrodynamic phenomena. It will now be seen
that it is possible to keep it in the search for the solution.
Because of the peculiar form of the source vector, namely its dependence on the gradient of a
primitive variable, Yb, system (4.87), can be written in a standard quasi-linear, autonomous,
non-conservative form
( ) ( )t x∂ + ∂ =V V 0A B (4.92)
where
( )= ∂ +V F *B A (4.93)
346
and A* is such that bf=*
23A and 0=*ijA for all other values of i and j. The coefficients of
the matrices B and A are shown in Annex 4.2. In order to keep the equations comparable to
those of previous works (eg. de Vries 1965, Lyn 1987, Sloff 1993, Morris & Williams 1996),
the second line of A is pivoted to eliminate the coefficient of ( )t h∂ .
Thus, matrix B is not a Jacobian matrix. Furthermore, it follows from (4.93) and (4.91) that it
is not possible to write ( )b x bf Y∂ in conservative form. If that was possible, A* would be a
Jacobian matrix of the transformation between ( )δ U and
( )δ V . But this is impossible
because, in what concerns (4.92), the relation between ( )δ U and
( )δ V is already specified
by A.
The fact that (4.92) has no true conservative form is of little importance for the purpose of
this section. In fact, it can be shown that the system (4.92) has the essential property of a
conservative system; equation (4.90) is valid.
Proposition 3.1
Let ] [ 2( , ) : 0,h u × +∞ → be piecewise continuous with compact support,
fb(h(x,t),u(x,t)) smooth and of bounded variation and Yb(x,t) Lipschitz continuous a.e. with
compact support. If Yb is independent of (u,h), then
( ) *2db bf Y M
+∞
ξ−∞
∂ ξ =∫
where *2M is independent of t.
The proof of the above proposition is direct. The space integration of ( )b x bf Y∂ will be first
carried out and integration by parts applied
( ) ( )d lim dm
b b b bmm
f Y f Y+∞
ξ ξ→+∞
−∞ −
∂ ξ = ∂ ξ =∫ ∫
( ) ( ) ( ) ( ) ( )lim , ( , ), ( , ) , ( , ), ( , ) dm
b b b b b bmm
Y m t f h m t u m t Y m t f h m t u m t Y fξ→+∞
−
⎧ ⎫⎪ ⎪− − − − − ∂ ξ⎨ ⎬⎪ ⎪⎩ ⎭∫
Since Yb is of compact support, it can be assumed that it vanishes outside arbitrarily large
[ ],a b ∈ and
( ) ( )d db
b b b ba
f Y f Y+∞
ξ ξ−∞
∂ ξ = ∂ ξ =∫ ∫ ( ) ( )lim d dm b
b b b bmm a
Y f Y fξ ξ→+∞
−
− ∂ ξ = − ∂ ξ∫ ∫
From Riez representation theorem (Giles 2000, p. 107), for which the premises of this
proposition are necessary, the latter integral exists. Because the primitive variables are
independent among themselves, or, at least, Yb is independent of (u,h), b bf Y≠ and
( ) ( )x b x bf Y∂ ≠ ∂ , wherever the derivatives exist. Under these conditions,
347
( ) ( )d db b
b b b ba a
f Y Y fξ ξ∂ ξ = − ∂ ξ∫ ∫
only if the integrand does not depend on h, u and Yb. It is easy to see that this is equivalent to
time invariance, since the integrand varies with t only through h, u and Yb.
It follows from Proposition 3.1 that (4.90) is true for system (4.92) provided that the premises
are true. The verification that fb is of bounded variation is elemental; this function can be
expressed as a difference of two monotone functions (see §3.1, equation (4.89)). The
Lipschitz continuity of Yb will be verified by computing the solution. It can be advanced that
Yb is piecewise C1 in its domain and that the derivative is indeed bounded.
As a final remark, it is intuitive that G affects the shape of the domains of dependence and
influence of a given point in the domain of V, as G depends on the gradient of a primitive
variable. Thus, the computation of the characteristics of the system should include bf . Thus,
it is utterly desirable to keep ( )b x bf Y∂ in the system of conservation laws. The shortcomings
of this approach will be explained later.
4.2.4 Eigenstructure and monotonicity
The nature of (4.92) is investigated through the characteristic polynomial 0− λ =B A , where
the operator ⋅ stands for the determinant operator. The characteristic polynomial can be
written as
3 2
1 2 3 0a a aλ + λ + λ + = (4.94)
where the coefficients a1, a2 and a3 depend on u and h and on its derivatives and are required
to be ( )1C . The expressions for these coefficients are shown in Annex 4.2. It is now clear
why it was necessary to introduce the approximations referred to in §4.3.2.2, p. 343: since a1,
a2 and a3 depend on the derivatives of the closures (4.79), (4.80) and (4.81) the latter must be
( )1C with respect to h and u so that the former are ( )1C .
The characteristic polynomial (4.94) possesses three real and different roots, such that
λ(1)>λ(2)>λ(3), called the characteristics of the system. Consequently, there are three
independent eigenvalues associated to each of the λ(k) eigenvalues which form the base of
eigenspace whose dimension is equal to the rank of − λB A . It is thus concluded that the
system is strictly hyperbolic (cf. Witham 1974, p. 116). The characteristics are functions of u
and h and can be plotted as a function of the Froude number. Figure 4.20(a) shows the
characteristics corresponding to system (4.76) to (4.78). It is shown that the influence of the
sediment inertia does not change the structure of the characteristic fields, namely that
λ(1)>λ(2)>0 and that λ(3) is always negative. Figure 4.20(a), relative to the system (4.73), (4.74)
and (4.75), shows that the characteristic fields are also λ(1)>λ(2)>0 and that λ(3)<0. The
magnitude of the characteristics depends on the mobility of the sediment. Increasing the
mobility of the sediment by changing the internal friction angle and the density of the
sediment, it is observed that all the characteristic fields most affected are affected those
corresponding to λ(2) and λ(3). However, the structure of the fields and the order of magnitude
remains the same.
348
-1.0
0.0
1.0
2.0
3.0
0.0 0.5 1.0 1.5 2.0Fr (−)
/(gh
0 )0.
5 (-
)
-1.0
0.0
1.0
2.0
3.0
0.0 0.5 1.0 1.5 2.0Fr (−)
/(gh
0 )0.
5 (-)
FIGURE 4.20. Eigenvalues of (4.92) as a function of the Froude number. a) Influence of the
sediment density; red line ( ) stands for the characteristics of system (4.76) to
(4.78) with (s − 1) = 0 (A = 0) and blue line ( ) stands for the characteristics of the
same system with s = 2.65. b) Influence of sediment mobility; blue line ( ) stands
for tan(ϕb) = 0.7, s = 2.65, red line ( ) and stands for tan(ϕb) = 0.45, s = 1.65.
The remaining analysis is carried out with the characteristics of system (4.73), (4.74) and
(4.75). The expressions for each of the λ(k) are too long to be meaningful. It is preferable to
plot the values of λ(k) within the physical domain of V. The results are shown in figures 4.21,
4.22, 4.23 and 4.24.
The physical domain of V (the space of the dependent variables) can be reduced to a square
of + +× , since the entries of B and A, and thus of λ(k), are functions of h and u, and it is
assumed that the solution will feature non-negative velocities and water depths. The upper
bounds of the square are determined from the initial conditions. The maximum water depth is
the initial water depth in the reservoir h0 = YL – YbL (figure 4.1). It is expected that the
maximum flow velocity for a geomorphic dam-break flow does not exceed the velocity of the
wave-front obtained under the ideal conditions of the Ritter solution, namely, clear water and
fixed smooth bed (Stoker 1958, p. 340). It can be shown that this is the case for the
horizontal bed problem, but it is not necessarily under the more general initial conditions
shown in figure 4.1, namely YbL > YbR. Lacking a better estimate, it will thus be admitted that
the upper bound for the velocity is 0 02u gh= .
Thus, the space of the dependent variables is the bounded interval
( ){ }0 0, : 0 0 2hu h u h h u gh+ +Δ = ∈ × < ≤ ∧ < ≤ .
Any solution will be represented by a line in huΔ . The remaining dependent variable, Yb, is
completely determined in that line because, as it will be seen, the solution is piecewise
monotone and self-similar. Thus, the transformations ( )hξ ξ and ( )uξ ξ , /x tξ = are
injective and it is always possible to find transformations ( )1( ) bu u Y−ξ = ξ or
( )1( ) bh h Y−ξ = ξ .
349
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2
0
2
4
lH1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2
0
2
4
lH2LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2
0
2
4
lH3LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
c)
FIGURE 4.21. Surface and density plots of ( )
0k ghλ . a) k = 1; b) k = 2; c) k = 3.
Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
-2
-101234
0.33
0.67
1
1.33
1.67
2
0.67
1.332
0
ugh
0
hh
0
hh
0.33
0.67
1
1.33
1.67
2
0.33
0.67
1
1.33
1.67
2
0
hh
0
hh
0.67
1.332
0
ugh
0
hh
0.67
1.332
0
ugh
0
hh
350
00.25
0.50.75
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
02
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2
0
2
4
lH1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
00.25
0.50.75
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0
FIGURE 4.22. Surface plot of ( )
0k ghλ . It is shown that the system is always stricktly
hyperbolic since λ(1)>λ(2)>λ(3) in all of huΔ . Computations performed with Cf0 = 0.1,
tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Figures 4.21, surface and density plots, and 4.22, a comparative surface plot, show the
behaviour of each λ(k)(u,h) regarding signal and monotonicity. In particular, it is shown in
figure 4.22 that the characteristics are strictly λ(1)>λ(2)>λ(3), throughout the domain. Figures
4.23, 4.24 and 4.25 are solely concerned with monotonicity, as they show the gradients of λ(k)
regarding h and u.
The first characteristic field is always positive in huΔ and it is simultaneously monotone in
both u and h (figures 4.21(a) and 4.23) except in a narrow strip near h = 0. This is easily seen
in figures 4.23(a) and 4.23(b). The u-derivative is greater than zero in huΔ but, near h = 0,
( )(1)u∂ λ appear to be smaller than zero. Varying Cf0, and thus the intensity of the sediment
transport, it is observed that this narrow region does not widen.
Except for the strip near h = 0, whose importance is neglected because it does not vary with
the sediment transport parameters, the first characteristic field is essentially monotone,
according to the following definition.
Definition 3.1: 2:F → is essentially monotone iff, for every 2( , ), ( , )A A B BA h u B h u= ∈ , there is at least one bijection ] ] ] ]0 0: 0, 0,u hφ → such that
( ) ( )0A B F B F A− > ⇒ ≥
or
( ) ( )0A B F B F A− > ⇒ ≤
for a given norm, and ( )A Au h= φ and ( )B Bu h= φ .
-2
-1
0
1
2
3
4Scale
0.67
1.332
0
ugh
0
hh
351
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2-1012
$%%%%%%%%h0ÅÅÅÅÅÅÅg ∑hHlH1LL
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2-10
1
2
∑uHlH1LL
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
FIGURE 4.23. Surface and density plots of: (a) ( )(1) 0h
hg
∂ λ and (b) ( )(1)
u∂ λ . (1)λ is a
essentially monotone characteristic field except in a narrow strip near h = 0.
Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
From the above definition it is clear that a function defined in a sub-interval of 2
is
essentially monotone if it does not possess any crest-like or valley-like lines, i.e., lines of
zero gradient in some direction. Without loss of generality, the norm may be the Euclidean
2– norm. This is a necessary condition for non-linearity. Thus, it can be anticipated that the
λ(1)-field is genuinely non-linear. Genuinely non-linear fields assume as solutions only
shocks or rarefaction waves, contact discontinuities are precluded. Hence, it can also be
anticipated that, as in the clear water shallow water equations, the solution for λ(1)-field is
expected to be a shock.
-2
-1012
0.33
0.67
1
1.33
1.67
2
0
hh
1.33
0
ugh
0
hh 0.67
2
0.33
0.67
1
1.33
1.67
2
0
hh
1.33
0
ugh
0
hh 0.67
2
352
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2-1012
$%%%%%%%%h0ÅÅÅÅÅÅÅg ∑hHlH3LL
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2-10
1
2
∑uHlH3LL
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
FIGURE 4.24. Surface and density plots of: (a) ( )(2) 0h
hg
∂ λ and (b) ( )(2)
u∂ λ . (2)λ is a
essentially monotone characteristic field in huΔ . Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
The second characteristic field is such that the values of λ(2) are always positive and smaller
than the values of λ(1) in huΔ (figure 4.22). The derivatives of λ(2) are such that ( )(2) 0u∂ λ >
and ( )(2) 0h∂ λ < , as seen in figure 4.21(b) and, explicitly, in figure 4.24. Therefore, it is
clear that λ(2)-field is also essentially monotone. It is anticipated that there is a lack of
symmetry between the wave structure of the solution of the Riemann problem for the one-
dimensional Euler equations and the geomorphic shallow-water equations. It is expected that
the λ(2)-field is genuinely non-linear while the middle characteristic field for the Euler
equations is linearly degenerate (cf., eg, LeVeque 1990, pp. 89-93)
-2
-1012
0.33
0.67
1
1.33
1.67
2
0
hh
0.671.33
2
0
ugh
0
hh
0.33
0.67
1
1.33
1.67
2
0
hh
0.67
1.332
0
ugh
0
hh
353
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2-1012
$%%%%%%%%h0ÅÅÅÅÅÅÅg ∑hHlH3LL
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2-10
1
2
∑uHlH3LL
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
FIGURE 4.25. Surface and density plots of: (a) ( )(3) 0h
hg
∂ λ and (b) ( )(3)
u∂ λ , thick line
stands for ( )(3) 0u∂ λ = , hence (3)λ is not a strictly monotone characteristic field in
huΔ (see figure 10). Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and
ds = 0.003 m. Scale:
In the huΔ domain, the values of λ(3) are always negative, as seen in figures 4.21(c) and 4.22.
There is strict monotony with respect to h, i.e., ( )(3) 0h∂ λ < as easily seen in figures 4.21(c)
and 4.25(a), but not with respect to u. This is an important feature that can be observed
neither in figure 4.21(c) nor in figure 4.25(b). Detailed contour plot of figure 4.21(c) is
revealed in figure 4.26.
Figure 4.26 was computed with a much lighter sediment, s = 1.5, and with unreasonably high
values of the friction coefficient. In this case it used Cf0 = 0.3 in order to magnify the non-
monotonicity regarding u. Observing this figure one detects a crest line which separates a
-2
-1012
0.33
0.67
1
1.33
1.67
2
0
hh
0.671.33
2
0
ugh
0
hh
0.33
0.67
1
1.33
1.67
2
0
hh
0.67
1.332
0
ugh
0
hh
354
zone where ∂h(λ(3)) < 0 and ∂u(λ(3)) > 0 (lower part of the plot) from a zone where the field is
concave relatively to u, i.e., ∂h(λ(3))h < 0 and ∂u(λ(3)) < 0.
FIGURE 4.26. Detailed contour plot of (3)
0ghλ . It is shown that characteristic field is not
essentially monotone in all of huΔ . Thick line ( ) represents ∂u(λ(3)) = 0. Dotted
line ( ) represents Fr = 0.7. Computations performed with Cf0 = 0.3, tan(ϕb) = 0.5,
s = 1.5 and ds = 0.003 m.
Unlike the non-essentially monotone region in the λ(1)-field, the magnitude of the concave
region beyond the crest line revealed in figure 4.26 grows visibly with the intensity of
sediment transport. This is the reason why this apparently insignificant feature deserves this
amount of attention. Concave characteristic fields are not common in water flow problems and
arise, for instance, in oil-recovery problems (Isaacson 1986, Rhee et al. 1989, p. 63-72).
Formally, the existence of the crest line may signify that the field is not genuinely non-linear
in huΔ . In this case, both a shock and a rarefaction wave can, simultaneously, constitute the
Riemann solution for the λ(3)– field, as in the Buckley-Leverett flux equation. This is an
unlikely, although theoretically possible solution for a dam-break flow.
Being the only negative field, the third characteristic field must be responsible for the
connection between the undisturbed left state (figure 4.1) and a constant state in the vicinity
of the dam. A solution involving both a rarefaction wave and an upstream propagating shock,
that leaves a smaller water depth on its downstream side, would have no correspondence to
any observation done on dam-break flows with or without movable bed. It is in this sense that
this solution is unlikely.
In order to evaluate the degree of dependence of each of the characteristic fields with the
magnitude of the sediment transport, it is important to note that, as C → 0, the eigenvalues of
matrix B become the solutions of ( )( ) 0=−−λ+−λλ ghughu . This is easily shown by
taking the limit C→0 of the expressions of a1, a2 and a3 shown in the Annex 4.2. The
characteristics of the resulting shallow-water system are the well known
ghu +=λ≡λ +)1(, 0)2( =λ and ghu −=λ≡λ −)3(
. The relative magnitude of the
characteristics of the geomorphic shallow water equations are seen in figure 4.27. It is clear
from this figure that the first characteristic field is almost unaltered by the sediment phase.
On the contrary, the geomorphic variables deeply affect (3)λ and
(2)λ , especially for Froude
numbers between 0.7 and 1.5. The concave zone in figure 4.26 is likely to be sensitive to the
0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
h0.0 0.1 0.0
u
0.0
0.6
0.3
355
parameters that control the transport of sediment because it occurs near the lines that
represent that range of Froude numbers, as seen in figure 4.27.
4.2.5 Non-linearity
The study of the non-linearity of the characteristic fields requires the computation of the
right eigenvectors of matrix − λB A . These are obtained from
( )− λ =r 0B A (4.95)
Simple algebraic manipulations lead to the following expressions, as function of λ(k)
( )( ) ( ) ( )5 21 (1 )k k kr p h N N= − − + λ λ (4.96)
( )( )( ) ( ) ( )4 12 (1 ) (1 )k k kr p u N p N= − − + + − − λ λ (4.97)
( )( )( ) ( ) ( )4 5 5 1 23
k k kr N h N u N N h N u= − + − + − λ λ (4.98)
where the coefficients are those listed in Annex 4.2. The compatibility with the clear water
case must be ensured. It is easy to show that if the sediment concentrations and their
derivatives are zero, equations (4.96) to (4.98) become (1)
1 1r = , (1)2r gh h= ,
(1)3 0r = ,
(2)1 0r = ,
(2)2 0r = ,
(2)3 0r = ,
(3)1 1r = ,
(3)2r gh h= − and
(3)3 0r = . It is clear that
( )(1) (1)(1)1 2,r r=r and ( )(3) (3)(3)
1 2,r r=r are the eigenvectors of the 2x2 Jacobian matrix of the
shallow water equations. The second eigenvector simply states that the second characteristic
field degenerates into a linear field for which the solutions are stationary waves of zero
amplitude.
The non-linearity of the characteristic fields should be verified, at least for the second and
third characteristic fields, since the first field appears to be similar to the one of the shallow-
water equations for fixed bed. A characteristic field is genuinely non-linear if
( )( ) ( ) 0hu Yb
k k∈Δ ∪Δ∂ λ ≠ ∀V Vri (4.99)
where bYΔ is the domain for Yb, as generated by any of the remaining variables. The left-hand
side of (4.99) is the directional derivative of the characteristic in the direction of r.
It should be recalled that ( )
( )d
dk
kW=
Vr , where ( )kW is the characteristic variable
corresponding to the ( )kλ – field, such that ( ) ( )( ) ( ) ( ) 0k k k
t tW W∂ + λ ∂ = . Hence, condition
(4.99) ensures that across a k-wave, the path of the solution will not be overtaken by a line
of local maxima or minima of ( )kλ .
356
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-5
0
5
lH1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2
0
2
4
lH2LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-2
0
2
4
lH3LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
0
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
c)
FIGURE 4.27. Surface and density plots of: (a) (1)+λ λ , (b)
(2)−λ λ and (c) (3)−λ λ .
Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
-10-50510
0.33
0.67
1
1.33
1.67
2
0.67
1.332
0
ugh
0
hh
0
hh
0.33
0.67
1
1.33
1.67
2
0.33
0.67
1
1.33
1.67
2
0
hh
0
hh
0.671.33
2
0
ugh
0
hh
0.671.33
2
0
ugh
0
hh
(1)
+λ
λ
(3)
−λ
λ
(2)
−λ
λ
357
A pulse in a simple wave corresponding to a genuinely non-linear characteristic field
necessarily deforms. If the path of the solution is such that the characteristic is monotone
increasing, the pulse is self-sharpening and gives raise to a shock, even if the initial
conditions are smooth. If the characteristic is monotone decreasing, the simple wave is a
rarefaction wave. Thus, the only admissible waves in a non-linear characteristic field are
rarefaction waves or shocks; contact discontinuities are not admissible (cf. Rhee et al. 1989,
pp. 59-60).
It is well known that inequality (4.99) is verified for the fixed-bed shallow-water equations.
In order to verify that the same happens with the geomorphic shallow-water equations, a plot
of ( )( ) ( )k k∂ λV ri is shown in figure 4.28.
It is observed that condition (4.99) holds for both the first and the second characteristic
fields, as seen in figures 4.28(a) and (b). In fact, except for a limited region near h = 0 outside
figure 4.28(a), ( )(1) (1) 0∂ λ >V ri and ( )(1) (1) 0∂ λ <V ri .
As for the third characteristic field, it can be shown that essential monotonicity is a necessary
condition for non-linearity. Thus, the crest line seen in figure 4.25(b) and 4.26 indicates that
this field might have points where (4.99) does not hold. It is clear from figure 4.28(c) that
there is indeed a line in huΔ where ( )(3) (3) 0∂ λ =V ri .
Furthermore, a detailed analysis of (3)λ reveals that it changes its sign only once in huΔ .
Rhee et al. (1989), pp. 57-59, show that the (3)λ -characteristic field admits, as solutions,
rarefaction waves combined with shocks and name these solutions as semi-shocks. It is easy
to understand how a semi-shock can occur in the present (3)λ − field. Given that
(3)λ is
always negative, let it be imagined that the 3-wave connects a left constant state, with high h
and low u, and a right constant state with low h and high u. If the line ( )(3) (3) 0∂ λ =V ri
stands between the left and the right states, the 3-wave would span until that line is reached
and then fold back until the values of u and h on the right side are reached. This is described
in figure 4.29. Such a solution is not a good description of any observed laboratorial dam-
break flow (cf. Chapter 5, §5.2.5).
If ( )( ) ( )k k∂ λV ri is taken as a measure of the degree of the non-linearity of the field, it can
be concluded from figure 4.28 that the strength of the 1-waves is expected to be larger than
that of the 2-and 3- waves. For instance, if a shock develops as a solution of the second
characteristic field, its strength is expected to be lower than the strength of a shock in the
first characteristic field.
358
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
h0 ∑rH1L HlH1L LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
--
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
h0 ∑rH2L HlH2L LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
--
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
h0 ∑rH3L HlH3L LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
--
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅ ÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
c)
FIGURE 4.28. Surface and density plots of ( ) ( ) ( )
0 0( )k k kh ghΕ = ∂ λV ri : (a) k = 1, (b) k = 2,
(c) k = 3, thick line in the density plot of stands for (3) 0Ε = . Computations performed
with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
-4
-2024
0.33
0.67
1
1.33
1.67
2
0.671.33
2
0
ugh
0
hh
0
hh
0.33
0.67
1
1.33
1.67
2
0.33
0.67
1
1.33
1.67
2
0
hh
0
hh
0.671.33
2
0
ugh
0
hh
0.671.33
2
0
ugh
0
hh
359
FIGURE 4.29. Formation of a semi-shock as a solution for the third characteristic field. (a)
behaviour of the characteristic lines in the x-t domain. (b) qualitative behaviour of the
solution. Left and right states are on opposite sides of the line ( )(3) (3) 0∂ λ =V ri .
As final conclusion, it can surely be stated that none of the characteristic fields will develop
contact discontinuities, hence breaking the symmetry between the Euler equations and the
geomorphic shallow-water equations.
4.3 DESCRIPTION OF WAVES THAT CHARACTERIZE TO THE POSSIBLE RIEMANN
SOLUTIONS
4.3.1 Existence of solutions
A Riemann problem is the Cauchy problem
( ) ( )( )t x∂ + ∂ =U F U 0 (4.100)
L 0
0R 0
,( ,0)
,x x
xx x
≤⎧≡ = ⎨ >⎩
UU U
U (4.101)
where (4.100) is hyperbolic. Lax (1957) showed that the solution of the Riemann problem
exists, for a generic system of n equations, provided that: i) system (4.100) is strictly
t
x
maximum λ(3) shock
(a)
solution of h
x/t
hL
hR uL
uR
solution of u (b)
360
hyperbolic, ii) ( )1C n∈F (is Lipschitz continuous, see, e.g. Ambrosio et al. (2000), p. 12),
iii) all the characteristic fields are genuinely non-linear and iv) L R−U U is small. In that
case, the solution is composed of n+1 constant states separated by shocks or centred simple
waves (or rarefaction waves). All waves are, thus, centred at the origin, as seen in figure
4.30. Following Lax’s (1957) work, a large body of research on the existence of solutions for
(4.100) was built over the time. Less restrictive conditions were studied by Glimm (1965),
Smoller (1969) or Liu (1974), but most research was performed for systems of two equations.
FIGURE 4.30. General wave structure of the Riemann solution for the geomorphic dam-break
problem.
System (4.92) cannot, as seen in §4.2.3, be written in the form of (4.100). Also, it cannot be
ensured that dam-break problems feature small L R−U U . Thus, it makes sense to discuss
the existence of solutions of (4.92), subjected to (4.101). For that purpose, let (4.92) be
written as
( )( ) ( )( ) ( )t x x∂ + ∂ + ∂ =U V F V V 0*A
(4.102)
Dafermos (1973), working on a 2x2 system, proved the existence of solutions of (4.100) and
(4.101) by taking the limit 0ε → of ( ) ( ) ( )2t x xt∂ + ∂ = ε ∂U F U . This viscosity method can be
used to prove the existence of solutions of (4.102) and (4.101), where the initial data, U0, is
that of figure 4.1. First, it is noted that (4.102) is invariant under the transformation
( ) ( ) \0, , , ax t ax at+∈∀ , which is the same as asserting that the Riemann problem admits
self-similar solutions. Therefore, if x tω = , the system
( )( ) ( )( ) ( ) ( )2( , ) ( , ) ( , ) ( , )t x x xx t x t x t t x t∂ + ∂ + ∂ = ε ∂U V F V V V*A
is equivalent to
( )( )( ) ( )( )( ) ( ) ( )( )2( )ω ω ω ω−ω∂ ω + ∂ ω + ∂ ω = ε ∂ ωU V F V V V*A (4.103)
subjected to the initial data
L( )−∞ =U U , and R( )+∞ =U U (4.104)
In order to prove the existence of solutions for the geomorphic shallow water equations, the
following theorems will be invoked.
x
t
undisturbed L-state undisturbed R-state
wave associated to λ(1)
wave associated to λ(2)
wave associated to λ(3) constant state (2) constant state (1)
361
Theorem 4.1 (Dafermos 1973)
Assume that there is M, independent of m, such that every possible solution ( )ωU of
(4.103) subjected to L( )m− =U U and R( )m =U U m ≥ 1, satisfies
( )supm m− <ω<
ω <U M
Then, there exist solutions of (4.103) subjected to (4.104).
Theorem 4.2 (Dafermos 1973)
For a fixed 0ε > , let ( )ε ωU denote the solution of (4.103), subjected to (4.104). Suppose
that the set { }: 0 1ε < ε <U is of uniformly bounded variation. Then, there exists a
function ( )ωU of bounded variation such that ( )/x tU is a weak solution of (4.102)
subjected to (4.101).
If the conditions of Theorem 4.1 are met, then ( )ε ωU exists. It follows that the existence of
solutions for (4.102), (4.101) is dependent on the boundedness of the solution. In particular, it
is necessary to prove that the there is a sequence { }jε , such that 0j+ε → as j → ∞ , and a
function ( )ωU of bounded variation, such that ( ) ( )jε εω → ωU U , a.e. on ] [,−∞ +∞ as
j → ∞ . The results of Dafermos are applicable in this study provided that the unicity of the
solutions of (4.103) is proved. The following proposition must be proved true.
Proposition 4.1. Let the ( )∂V F and A* be continuous in uhΔ . For given 0ε > the Cauchy
problem for (4.103) has a unique solution for a class of initial conditions that comprises a
null-measured set of discontinuities.
If ( )∂ =V U I and if A* and ( )∂V F are Lipschitz continuous, proposition 4.1 would follow from
the general theory of ODEs (cf. Braun 1993, p. 129). Unfortunately, A* is not amenable to the
identity matrix and, since the derivatives of some of the closure equations are unbounded, a
proof must be constructed. For that purpose, let AV and
BV be two solutions of (4.103)
defined in some bounded interval centred in ω0 such that ( ) ( )A B0 0ω = ωV V and
( ) ( )A B0 0ω = ωV V
i i, where the overdot stands for derivative to ω. Defining
( ) A BD ( ) ( ) ( )ω ≡ ω − ωV V Vi i
, it is noted that, from the mean value theorem,
( ) ( ) ( )( )( )A B A B− = ∂ −V
F V F V F V V V (4.105)
Because the derivative at the mean point is a constant, it is true that
( ) ( ) ( )( ) ( )0
A B D ( ) dω
ω
− = ∂ ξ ξ∫ VF V F V F V V
362
On the other hand
( ) ( ) ( )( )( )A B A B− = ∂ −V
U V U V U V V V
( )( ) ( )( ) ( )( ) ( )A B Dω ω∂ − ∂ = ∂V
U V U V U V V (4.106)
If (4.102) is integrated
( )( )( ) ( )( )( ) ( ){ } ( )( )
0 0 0
2d + d dω ω ω
ω ω ω ωω ω ω
− ω∂ ω ω + ∂ ω ∂ ω = ε ∂ ω ω∫ ∫ ∫U V F V V VA*
( )( ) ( )( ) ( )( ) ( )( )( )0 0ω ωω − ω = ε ∂ ω − ∂ ωF V F V V V
( )( )( ) ( ){ }
0
dω
ω ωω
− ω∂ ω − ∂ ω∫ U V V*A (4.107)
From (4.105), (4.106) and (4.107)
( )( ) ( ) ( ) ( )( )0
A BD ( ) dω
ω ωω
∂ ξ ξ = ε ∂ − ∂∫ V F V V V V
( )( ) ( )( ) ( ) ( ){ }0
A B A B dω
ω ω ω ωω
⎡ ⎤ ⎡ ⎤− ω ∂ − ∂ − ∂ − ∂ ω⎢ ⎥ ⎣ ⎦⎣ ⎦∫ U V U V V V*A
( )( ) ( ) ( ) ( )( ) ( )0 0
D ( ) d D ( ) D dω ω
ω ω
⎡ ⎤∂ ξ ξ = ε ω − ω∂ − ω⎣ ⎦∫ ∫V V
F V V V U V V*A
( )( ) ( )( ) ( ) ( )0
D ( ) d D ( )ω
ω
⎡ ⎤∂ − ξ∂ + ξ ξ = ε ω⎣ ⎦∫ V V
F V U V V V*A (4.108)
since ( )( ) ( )( )∂ − ξ∂ +V V
F V U V *A is continuous, it follows from (4.108) and from
Gronwall’s inequality (Hirsch et al. 2004, p. 393), with K = 0, that ( )D ( )ω =V 0 and, hence,
A B=V V , which completes the proof.
Proposition 4.1 and Theorem 4.1 allow for the application of Theorem 4.2. Hence, it is proved
that there are indeed weak solutions of (4.102) subjected to (4.101).
The unicity of the weak solution must be addressed now. It should be recalled that a function
( , )x tU is a weak solution of (4.100) iff there exists a function :φ × → ,
[ [( )10 0,Cφ ∈ × +∞ , i.e., smooth and with compact support, such that
( ) ( )( ) 0 00
( ) d d dt x x t x+∞ +∞ +∞
−∞ −∞
∂ φ + ∂ φ = φ∫ ∫ ∫U F U U (4.109)
363
The derivatives of the potentially discontinuous ( , )x tV are transferred to the smooth φ and
conservation, thus allowing for the generalised definition of the solution. Hence, a weak
solution admits discontinuities in its domain provided the set of points where they occur is of
null measure. In other words, a weak solution is smooth almost everywhere.
Given that the characteristic fields are genuinely non-linear, the only discontinuities in the
profiles of the dependent variables are shock waves. Across the shock waves, equations
(4.100) or (4.102) are not valid because the derivatives do not exist. Only the integral form of
the conservation laws is valid. Hence, shock conditions, or Rankine-Hugoniot conditions, must
be derived from (4.109). A system of conservation laws written in conservative form admits
the shock relations
FU Δ=Δ S (4.110)
where ΔU = Udownstream – Uupstream , ΔF = Fdownstream – Fupstream and S is the velocity of the
shock.
Back to the problem of unicity, it is recalled that, in general, there can be more that one weak
solution for a given set of initial conditions. The works of Lax (1957) and Olga Oleinik, 1959
(cited by Magenes 1996) set the fundamental of the theory of the unicity of PDEs. In this
paper, the fundamental result is that unicity is ensured if Liu’s (1974) extended entropy
condition
( ) ( ) ( )L Rk k kSλ > > λ (4.111)
( 1) ( ) ( 1)R Lk k kS− +λ > > λ (4.112)
is met. In (4.111) and (4.112) S(k) stands for the shock velocity. Equation (4.111) states that,
in an entropy satisfying shock, the characteristic lines corresponding to two adjacent constant
states must converge into the shock path. Equation (4.112) ensures that a shock in a given
characteristic field is not overtaken by characteristics of a different type.
It was seen before, in §4.2.3, that (4.102) has the properties of a conservation law. Yet, since
the derivation of the Rankine-Hugoniot conditions from (4.109) involves the use of Green’s
theorem, the conservation laws must be written in divergence form. As seen in §3.1, this is
not, in general, possible for system (4.102). Hence, a mathematical trick must be employed:
matrix A* must be linearised across the discontinuities, i.e., it should be a combination of the
values of V at the left and right states adjacent to the discontinuity.
The shortcoming envisaged at the end of §3.1 is clear now: the solution of (4.102) is unique
but only for the chosen linearization. Furthermore, not all linearisations are admissible. If the
shock strength tends to zero, the chosen combination of left and right states must converge to
the expression of A* valid in smooth regions. It was shown that the generic unique Riemann
solution of (4.102) subjected to (4.101), has the generic structure shown in figure 4.30. The
particular structure corresponding to the expressions embedded in (4.102) will be discussed
next.
4.3.2 The structure of the Riemann solution
Given that the λ(k)-fields are such that λ(1) > λ(2) > λ(3)
, it is expected that the fastest wave,
separating the undisturbed right state (R-state) from the constant state (1), would be
364
associated to the λ(1)-field, as seen in figure 4.28. The wave associated to the λ(2)
-field ought
to be the middle wave, connecting constant states (1) and (2), and should travel downstream,
since λ(1) > λ(2) > 0 > λ(3). Associated to the only negative characteristic, λ(3)
is an upstream
moving wave, separating the undisturbed left state and the constant state (2).
Physically admissible shocks verify the condition (4.111). Rarefaction waves are simple
centred waves that connect two constant states through a smooth transition. A rarefaction
wave is graphically identified by a fan of characteristic lines whose bounding lines obey
( ) ( )L Rk kλ < λ (4.113)
Judging upon the monotonicity of the characteristic fields and considering the inequalities
(4.111) and (4.113), one can determine the zones in Δuh for which the solution is a rarefaction
wave or a shock wave.
Let the point (h0,u0), a generic point in an essentially monotone region of Δuh, represent the
constant state upstream from a given wave. It should be recalled that all of Δuh is essentially
monotone for the first and the second characteristic fields (figures 4.21(a) and (b) and figures
4.23 and 4.24). On the contrary, the third characteristic field presents a crest line (figures
4.25(b) and 4.26). Also, there is a line such that ( )(3) (3) (3) ( ) 0u h∂ λ = − κ =V ri . The region
CR(3) = {u,h : u > κ(3)(h)} will herein be called the concave sub region of Δuh .
From the density plots of figures 4.21, 4.23, 4.24 and 4.25, one can determine the sub-
regions of existence of shock and rarefaction waves for each of the characteristic fields as a
function of the location of the downstream point (h1,u1). Indeed, any sub-region for which
λ(k)(u0,h0) > λ(k)(u1,h1) is a region for which a shock is the admissible solution. It should be
noticed that this is a necessary condition and not a sufficient one since it is nowhere proved
that, in between those characteristics, there is a shock speed obeying inequality (4.111) and
(4.112). Contrarily, the set of all points for which λ(k)(u0,h0) < λ(k)(u1,h1) is a sub-region for
which the solution is, in accordance to (4.113), a rarefaction wave.
The results are shown in figures 4.31(a), (b) and (c). The origin of these plots is point (h0 , u0) = (0.20, 1.2), but the results are qualitatively valid for any other point. The thick line in
figures 4.29(a), (b) and (c) is the isoline that separates the shock from the rarefaction wave
region and it has the following equation
Ψ(k)(u,h) = λ(k)(h,u) – λ(k)(h0,u0) = 0
or, if it can be made explicit for u,
u = ψ(k)(h) (4.114)
The solution for the λ(1)-field can be a rarefaction wave if (h1,u1) belongs to RW(1) = {u,h : u
> ψ(1)(h)}. This region, shown in figure 4.31(a), includes the first quadrant of that plot, (u>u0 ∧ h>h0), the upper part of the second quadrant, (u>u0 ∧ h<h0), and the upper part of the
fourth quadrant, (u<u0 ∧ h>h0). A shock wave would be expected if (h1,u1) lies in SW(1) = {u,h : u < ψ(1)(h)}.
As explained above, the λ(1)-field is such that λ(1) > λ(2) > 0 > λ(3)
. It is, thus, expected that
the constant state downstream of this wave is the undisturbed initial right state or R-state
(see figures 4.1 and 4.30), for which the velocity is typically zero (or very small) and the
365
water depth is typically small. The most likely situation in a dam-break flow is, thus, the
downstream point (h1,u1) lying on the third quadrant (u<u0 ∧ h<h0) in figure 15(a). The
expected solution for the λ(1)-field is, therefore, a shock wave.
-0.5
-0.25
0
0.25
0.5
-0.5 -0.25 0 0.25 0.5
(h - h 0)/h L
(u -
u0)
/(g h
L)0.
5
u > u 0
h > h 0
u < u 0
h > h 0
u > u 0
h < h 0
u < u 0
h < h 0
Shock-wave
Rarefaction wave
a)
-0.5
-0.25
0
0.25
0.5
-0.5 -0.25 0 0.25 0.5
(h - h 0)/h L
(u -
u0)
/(g h
L)0.
5
u > u 0
h > h 0
u < u 0
h > h 0
u > u 0
h < h 0
u < u 0
h < h 0
Shock-wave
Rarefaction wave
b)
-0.5
-0.25
0
0.25
0.5
-0.5 -0.25 0 0.25 0.5
(h - h 0)/h L
(u -
u0)
/(g h
L)0.
5
u > u 0
h > h 0
u < u 0
h > h 0
u > u 0
h < h 0
u < u 0
h < h 0
Shock-wave
Rarefaction wave
Concaveregion
c)
FIGURE 4.31. Regions of possibility for shock- or rarefaction waves for each of the λ(k)-fields.
a) k = 1; b) k = 2; c) k = 3. Computations performed with Cf0 = 0.3, tan(ϕb) = 0.5, s = 1.5
and ds = 0.003 m.
366
The solution for the λ(2)-field can be a rarefaction wave, if (h1,u1) belongs to RW(2) = {u,h : u
> ψ(2)(h)}, or a shock wave, if (h1,u1) belongs to SW(2) = {u,h : u < ψ(2)(h)} (see figure
4.31(b)). These results were determined with the aid of the discussion on the monotony of the
field, based on figures 4.21(c) and 4.24. Being the middle wave (figure 4.30), it is hard to
infer the type of wave if one possesses only information about the initial state and the form of
the closure equations.
This issue will be addressed in detail later, in the certainty that the particular shape of
equation (4.114), k=2, is determined by the initial conditions and by parameters of the closure
equations. For the purposes of determining the structure of the Riemann solution, one should
expect either a shock or a rarefaction wave as a solution for the λ(2)-field.
A concave sub-region appears in Δuh for the λ(3)-field (figures 4.21(c), 4.25 and 4.26).
Outside this region, the solution can be either a rarefaction wave or a shock wave. In the first
case, (h1,u1) would belong to RW(3) = {u,h : u > ψ(3)(h)} whereas, in the second case, (h1,u1) would belong to SW(3) = {u,h : u > ψ(3)(h)}.
The depth averaged velocity in a typical reservoir is zero. The water depth measured at the
reservoir is necessarily the larger in the system (figure 4.1). The wave solution for the λ(3)-
field will, thus, separate the constant L-state (upstream, see figures 4.1 and 4.30) from the
constant state (2). Relatively to the former, the latter state is characterised by a larger flow
velocity and a smaller water depth, i.e., quadrant 2 in figure 4.31(c). As seen in that figure,
and under the hypothesis that (h1,u1) lies in the essentially monotone region of Δuh, the
solution for the λ(3)-field will be a rarefaction wave.
If, on the contrary, (h1,u1) belongs to the concave region of uh+, the solution comprises an
expansion wave and a shock wave. The velocity of shock wave is determined from the
considerations explained in §4.3.1 and illustrated in figure 4.29. Further attention will be
given to this matter in the next chapter.
After ruling out some of the wave combinations, according to the reasoning explained above,
figure 4.30 can be updated. The structures of the possible Riemann solutions for the
geomorphic dam-break problem are depicted in figure 4.32. The conditions for which each of
the wave-structures occur will be discussed in the next section.
4.3.3 Mathematical treatment of shock and rarefaction waves
Once the structure of the solution is known, there remains the problem of computing the
values of u, h and Yb in the constant states and across the rarefaction waves and the values
of the shock velocities.
Let u1, h1, Yb1 and u2, h2, Yb2 be the values of the primitive variables in the constant states (1)
and (2), respectively. Let S1 be the velocity of the shock-wave corresponding to the λ(1)–
field and, if that is the case, let S2 be the velocity of the λ(2)– field.
The problem admits 8 unknowns if the structure of the solution is of type A (see figures 4.32a
or 4.23). If it is pf type B (figure 4.30b) the problem admits 7 unknowns. If the solution for the
λ(3)-field is both a shock and a rarefaction wave, the velocity of that shock constitutes an
extra unknown.
367
FIGURE 4.32. Possible wave structures of the Riemann solution for the geomorphic dam-break
problem. Solution for λ(2)-field: (a) shock wave, type A; (b) rarefaction wave, type B.
In order to determine the solution for a particular Riemann problem, it is sufficient to use the
relations that involve the unknown quantities of the constant states upstream and downstream
which, according to Lax’s (1957) theorem, bound a given wave. Across a rarefaction wave,
the Riemann invariants (Prasad 2001, pp. 84-86) are computed from the differential equations
31 2( ) ( ) ( )
1 2 3
dd dk k k
VV Vr r r
= = (4.115)
Equation (4.115) can be cast in a 2x2 system of ODE’s
( )( )
( )2( )
1
d , ;d
kk
ukru f u h
h r= = ⋅ (4.116)
( )( )
( )3( )
1
d, ;
d
kkb
zkrY
f u hh r
= = ⋅ (4.117)
subjected to (ui , hi) and (Yb i , hi), and where
( ) ( )
4 1( )( )
5 2
(1 ) (1 )
(1 )
kk
u k
p u N p Nf
p h N N
− − + + − − λ=
− − + λ (4.118)
x
t
undisturbed L-state undisturbed R-state
shock associated to λ(1)
shock associated to λ(2)rarefaction wave associated to λ(3)
constant state (2)
constant state (1)
x
t
undisturbed L-state undisturbed R-state
shock associated to λ(1)
rarefaction wave associated to λ(2)
rarefaction wave associated to λ(3)
constant state (2) constant
state (1)
368
( )( )
( )( ) ( )
4 5 5 1 2( )( ) ( )
5 2(1 )
k kk
z k k
N h N u N N h N uf
p h N N
− + − + − λ λ=
− − + λ λ (4.119)
The coefficients in (4.118) and (4.119) are identified in Annex 4.2. From (4.116) to (4.119) it
is clear that u and Yb change across the rarefaction wave whenever f (k)u and f (k)
z are
different from zero.
As for the shock waves, jump discontinuities in the profiles of the dependent variables, the
Rankine-Hugoniot conditions (4.110) apply. It should be reminded that the governing
equations could not be cast in divergence form. As stated before, a local, across the
discontinuity, linearization is required for the term related to the gravity force in the
momentum conservation equation.
A control volume analysis of the discontinuity renders the following integral momentum
equation
( ) ( ) ( )( ) ( ) ( )( )R
L
2 2dx
tx
R S R x R x F x F x+ − + −∂ ξ + − − −∫
( ) ( ) ( )( ), 0b bY x Y x+ ++ −−Φ − =V V (4.120)
where 2:Φ → is a continuous function of the flow depth and velocity at the left and the
right states. Taking the limit as (x2 – x1)→ 0 and x2 → x +; x1 → x – one obtains
2S R FΔ − Δ ( ) ( ) ( )( ), 0b bY x Y x+ ++ −−Φ − =V V (4.121)
which is the generalised form of (4.110) for the linearization Φ . The function Φ is chosen to
ensure the compatibility between the integral and the differential formulae in smooth regions.
In smooth regions it is required that
( )
( ) ( ) ( ) ( )2
2 11
0 2 1
1lim , d , ;x
b b x bx xx
Y f h u Yx x
+ −ξ
− →Φ ∂ ξ = ⋅ ∂
− ∫V V
If the limit exists then ( )
( ) ( )2 1 0lim , , ;bx x
f h u+ −
− →Φ = ⋅V V , which is the only mathematical
requirement that Φ must obey. There is a physical constrain to the choice of Φ: the
mechanical energy of flow must decrease across a discontinuity. Possible choices for this
function are
( )( )
, ; if / >
, ; if / <
b
b
f h u x t S
f h u x t S
+ +
− −
⎧ ⋅⎪Φ = ⎨⋅⎪⎩
or the arithmetic mean
( ) ( ){ }12 , ; , ;b bf h u f h u+ + − −Φ = ⋅ + ⋅
The latter formulation is used throughout this work.
369
It is now possible to build the system of equations necessary to find the Riemann solutions
under investigation (figure 4.32). The system corresponding to the 8x8 problem depicted in
figure 4.32(a), i.e., two shocks and one rarefaction wave, is composed of
( ) ( ) ( )R 1 1 R 1Y Y S uh uh− = − (4.122)
( ) ( ) ( ) ( ) ( )( )2 2 2 21R 1 1 2R 1 R 1m m m mR R S u h u h g h h− = ρ − ρ + ρ − ρ
( ) ( )( )( )1R 12 R 1m m b bg h h Y Y+ ρ + ρ − (4.123)
( ) ( ) ( )R 1 1 R 1b bY Y S Cuh Cuh− = − (4.124)
( ) ( ) ( )1 2 2 1 2Y Y S uh uh− = − (4.125)
( ) ( ) ( ) ( ) ( )( )2 2 2 211 2 2 21 2 1 2m m m mR R S u h u h g h h− = ρ − ρ + ρ − ρ
( ) ( )( )( )11 22 1 2m m b bg h h Y Y+ ρ + ρ − (4.126)
( ) ( ) ( )1 2 2 1 2b bY Y S Cuh Cuh− = − (4.127)
( )(3)2 2
d , ;d uu f u hh
= ⋅ (4.128)
( )(3)2 2
d, ;
db
zY
f u hh
= ⋅ (4.129)
Equations (4.122) to (4.124) are the RH conditions across the shock associated to λ(1) while
equations (4.125) to (4.127) represent the jump conditions for the shock associated to λ(2).
Equations (4.128) and (4.129) are obtained from the Riemann invariants across the rarefaction
wave associated to λ(3). Since the general form of both f (k)
u and f (k)z, equations (4.118) and
(4.119), in (4.128) and (4.129) is highly non-linear, those equations are not, in general,
amenable to an analytic solution. Thus, the solution of (4.128) and (4.129) implies the
numerical computation of these ODEs from the boundary condition u0 = uL, h0 = hL and Yb0 = YbL, h0 = hL to the point h = h2.
The system of equations necessary to determine the 7x7 problem of figure 4.32b), i.e., two
rarefaction waves and one shock wave, is composed of equations (4.122) to (4.124), (4.128),
(4.129) and
( )(2)1 1
d , ;d uu f u hh
= ⋅ (4.130)
( )(2)1 1
d, ;
db
zY
f u hh
= ⋅ (4.131)
The conditions across the second rarefaction wave, associated to the λ(2)-field, are
expressed in (4.130) and (4.131). These require the boundary conditions u0 = u2, h0 = h2 and
Yb0 = Yb2, h0 = h2 to the point h = h1.
The proprieties of equations (4.118) and (4.119) will be discussed next. Across the
rarefaction waves of figure 4.32, the variation of the primitive variables h, u and Yb is
370
necessarily smooth and monotone. Thus, two questions arise: i) do (4.118) and (4.119) ensure
the monotone variation of h, u and Yb in all of Δuh and ii) what are the implications of the
singularity achieved by zeroing the denominator of (4.118) and (4.119), whose equation is
( ) ( )
5 2(1 )k kp h N Nβ = − − + λ (4.132)
Both questions are answered by looking at figures 4.33, 4.34, 4.35 and 4.36. Figures 4.33 and
4.34 show surface and density plots of the variation of (4.118) and (4.119) for each of the
λ(2)- and λ(3)
-fields. Figure 4.35(a) show that β(2) is always positive, which means that (4.118)
and (4.119) have no discontinuity for k = 2. On the contrary, it is clear from figure 4.35(b)
that there is a line for which β(3) = 0.
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
h0 r2H2LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
--
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
r3H2L
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
FIGURE 4.33. Surface and density plots of non-dimensional (a) (2)
uf , (b) (2)zf .Computations
performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
Figure 4.34(a) shows that f (3)u is negative in a sub-region of Δuh bounded by the line β(3) = 0.
At that line the denominator of (4.118) is zero and f (3)u is infinitely large. Approaching β(3) = 0
-4
-2024
0.33
0.67
1
1.33
1.67
2
0
hh
0.671.33
2
0
ugh
0
hh
0.33
0.67
1
1.33
1.67
2
0
hh
0.67
1.332
0
ugh
0
hh
371
from the right, f (3)u tends to infinity over negative values whereas approaching it from the
left, f (3)u tends to infinity over positive values.
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
h0 r2H3LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
0.20.4
0.60.8
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
--
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-4-2024
r3H3L
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
b)
FIGURE 4.34. Surface and density plots of non-dimensional (a) (3)
uf , (b) (3)zf .Computations
performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
Scale:
Let D(3) be the sub-region of Δuh on the “right-hand side” of the line defined by β(3) = 0
(figure 4.36). Upstream from the expansion wave associated to λ(3) one has the point h/h0 = H
= 1, u/(g h0)0.5 = U = 0 (see figure 4.32), clearly lying on D(3) (see figure 4.36). The
downstream point cannot be found outside D(3) because the solution of (4.128) or (4.129),
with boundary conditions defined on D(3), is valid only in that sub-domain. D(3)
is thus the
domain of occurrence of the wave associated to λ(3).
Given that f (3)u is negative in D(3)
, the velocity increases as the water depth decreases, the
flow accelerates from the undisturbed left state to the second constant state, as expected (cf.
Stoker 1958, p. 333). As for f (3)z, its values are positive in D(3)
(figure 4.34(b)), which means
that, along the λ(3) expansion wave, the bed elevation decreases with decreasing water depth.
-4
-2024
0.33
0.67
1
1.33
1.67
2
0
hh
0.671.33
2
0
ugh
0
hh
0.33
0.67
1
1.33
1.67
2
0
hh
1.332
0
ugh
0
hh 0.67
372
This result is in agreement to the most elementary physical intuition and also with the results
of Fraccarollo & Capart (2002).
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-1
-0.5
0
0.5
1
bH2LÅÅÅÅÅÅÅÅÅÅÅÅh0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
a)
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0 2
46
uÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!g h0
-1
-0.5
0
0.5
1
bH3LÅÅÅÅÅÅÅÅÅÅÅÅh0
0.20.4
0.60.8
1
hÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!h0
-
-
b)
FIGURE 4.35. Surface plot of β(k). (a) k = 2; (b) k = 3. Computations performed with Cf0 = 0.1,
tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
hÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!
h0
uÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅè!!!!!!!
g h0
FIGURE 4.36. Density plot of ( ) ( ) ( )(3) (3) (3) (3) (3) 0bh u u Y zf f∂ λ + ∂ λ + ∂ λ = . The discontinuity
corresponds to (3) 0β = . Thick line ( ) corresponds to ( )(3) (3) 0∂ λ =V r .
Computations performed with Cf0 = 0.1, tan(ϕb) = 0.5, s = 2.65 and ds = 0.003 m.
It was earlier identified that the solution for the λ(3)-field could be composed of an expansion
wave and a shock wave if the downstream point would lie in the concave region of Δuh (see
figures 4.25(c), 4.29 and 4.31). It is now clear that such a solution is possible only if D(3)
contains the essentially monotone region. Otherwise, the crest line defined by ∂u(λ(3)) = 0 lies
outside D(3) and λ(3)
becomes essentially monotone in this domain. Hence, the only admissible
solutions are rarefaction waves. Observing figure 4.35 it is clear that the crest line defined by
∂u(λ(3)) = 0 does indeed lie outside D(3). The λ(3)
-field is, thus, essentially monotone in its
domain and the only admissible solution is a simple rarefaction wave.
(3) 0β =
1.332
0
ugh
0
hh 0.67
1.332
0
ugh
0
hh 0.67
0.33
0.67
1
1.33
1.67
2
0
hh
373
As for the expansion wave associated to λ(2), figure 4.33 shows that both f (2)
u and f (2)z are
negative throughout Δuh. Since it is strictly β(2) > 0, as seen in figure 4.34(a), the solution of a
centred simple rarefaction wave is smooth throughout Δuh. In other words, the domain D(2) of
occurrence of the expansion wave associated to λ(2) is the totality of Δuh.
Before trying to solve (4.122) to (4.129), it should be recalled that the waves associated to
the λ(1)- and the λ(2)
-fields can be shock waves because λ(k)upstream > λ(k)
downstream (k=1,2,
figure 4.32). This last condition is a necessary one; the necessary and sufficient conditions
would be the fulfilment of the entropy conditions (4.111) and (4.112) for shock obeying the
RH conditions (4.110). It will be assumed that the necessary and sufficient conditions are met
and, following the computation of the solution, it will be shown that (4.111) and (4.112) were
indeed true.
4.4 SOLUTIONS OF THE GEOMORPHIC DAM-BREAK PROBLEM
The computational procedure depends on whether the solution is composed of one expansion
wave and two shocks (type A solution, figure 4.32(a)) or two expansion waves and one shock
(type B solution, figure 4.32b)). Latter it will be discussed what are the domains of validity of
each solution and what are the appropriate parameters to express it.
Assuming that the solution comprises an expansion wave in the 3rd characteristic field and
shock waves in the 1st and 2nd ones (solution of type A), the solution procedure can be
described as follows.
The variation of u and Yb across the maximum possible expansion wave associated to λ(3) are
computed first, as a function of h. The undisturbed upstream left state provides the boundary
conditions for the ODE’s (4.128) and (4.129). These equations are simultaneously solved until
the water depth, and corresponding velocity, approach a point (hβ,uβ) along the singularity β(3) = 0. The numerical solution of the ODE’s was achieved through a 4th order Runge-Kutta
scheme. The values of h, u and Yb along the expansion wave are stored in a database.
A trial value for h2, to which corresponds a value of u2 and Yb2, is chosen from the database.
Equations (4.122) to (4.127) are then used to compute h1, u1, Yb1 (variables in the constant
state (1)), S2 and S1 (velocities of the shocks). It should be noticed that at this stage there are
6 equations but only 5 quantities to be computed, since the choice of h2 implies choosing also
u2 and Yb2. One of the equations should be used to compute the error associated to a
particular h2. The error should be brought to zero by choosing a different h2. This shooting
method can be rationalised and refined to save computational time.
The solution composed of expansion waves in the 3rd and 2nd characteristic fields and a shock
wave associated to the λ(1)– field (solution of type B) was determined from the following
procedure.
As it is the case with the solution of type A, the variation of u, Yb and h across the expansion
wave associated to λ(3) are computed first, through a procedure identical, in every step, to the
one described above. The resulting values are stored in a database.
Trial values for h2, u2 and Yb2, are chosen from the database. These values provide the initial
conditions for the ODE’s (4.130) and (4.131). These ODE’s are numerically discretized and
solved by the same 4th order Runge-Kutta scheme until a given trial value of h1 is reached.
374
Associated to this value of h1 one obtains also the values of u1 and Yb1. The remaining
unknown is the shock velocity S1. There are three independent equations, (4.125), (4.126) or
(4.127), to compute this parameter. One of these can be used to compute S1 while the
remaining must be kept to evaluate the error of the iterative process.
New trial values of h2 and h1 should then be tested until the values of S1 are identical
irrespectively of the equation used. Again, this shooting method should be refined to minimise
the time involved in the computations.
To illustrate the procedures explained above, two computation examples are given. The
equations are solved for two different sets of initial conditions and types of sediment.
Solutions of types A and B where computed with pumice and sand as bed material,
respectively. The parameters for the closure equations and the values of the initial states are
displayed in table 4.1.
TABLE 4.1. Parameters of the examples of computation
Solution hL
(m) hR
(m) YbL
(m) Cf0
(-) ds
(m) s
(-) Type A 0.4 0.1 0.08 0.1 0.003 2.65 Type B 0.4 0.00004 0.08 0.1 0.003 1.5
Figure 4.37 shows the structure of the solution of type A. The corresponding profiles of the
primitive variables are shown in figure 4.38.
As stated before, the solution for the λ(1)-field is a shock wave. Characteristic lines converge
into the shock path, as seen in figure 4.37(a), confirming that this shock satisfies the entropy
condition (4.111). Across the rarefaction wave the λ(1)-characteristics bend (1st derivative
discontinuous) as their value increases as a result of the increasing velocity. Across the
shock associated to the 2nd characteristic field there is a small change of direction due to the
decrease of the velocity field in the constant state (1). Entropy condition (4.112) is also
verified.
The solution for the λ(2)-field is, as seen in figure 4.37(b), a shock wave for the initial
conditions chosen. It is an entropy-satisfying shock as the λ(2) characteristic lines of constant
states (1) and (2) converge to the shock as required by (4.111). The value of the λ(2)
characteristics is zero in the upstream constant state (vertical characteristic lines) and
increases as the lines cross the rarefaction wave.
The values of λ(2) for this type of solution are very much dependent on the sediment transport
rate. It is through this characteristic field that the coupling between water and sediment is
most visibly expressed. In fact, the main difference between the clear water dam-break
problem and the movable bed problem lies in the existence of a second shock wave, a
positive jump in the flow velocity (like an hydraulic jump in the water phase) associated to an
agradational “sediment bore”. The 3rd characteristic field admits a rarefaction wave as a
solution as seen in figure 4.37(c). Characteristic lines are deflected as they pass the shock
paths as a result of the abrupt changes in the water depth and flow velocity.
Figure 4.38(a) shows the geometric configuration of the solution. The two downstream-
moving discontinuities and an upstream moving depression wave are clearly visible. The
375
dotted line represents the sediment transport layer of thickness hc. This layer was computed
considering (4.79) and (4.80).
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
-10 -5 0 5 10 15X '(-)
T ' (
-)
a)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
-10 -5 0 5 10 15X ' (-)
T ' (
-)
b)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
-10 -5 0 5 10 15X ' (-)
T ' (
-)
c)
FIGURE 4.37. Characteristic lines and Riemann wave structure for solution of type A. a) λ(1)-
field; b) λ(2)-field; and c) λ(3)
-field. Initial conditions and parameters of the closure
equation as shown in table 4.1.
376
Figure 4.38(b) shows the depth averaged flow velocity, u, the total discharge per unit width,
q, and the total sediment concentration, C. It is clear from figures 4.38(a) and 4.38 (b) that,
when hR is sufficiently large, the strongest sediment transport rate occurs at the constant
state (2) and not at the tip of the shock associated to the λ(1)-field.
-0.25
0.000.25
0.50
0.75
1.001.25
1.50
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5X ' (-)
Z' (
-)
a)
0.00
0.25
0.50
0.75
1.00
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5X ' (-)
U', Q
', C (-
)
b)
FIGURE 4.38. Example of solution of type A. The profiles are shown in a self-similar
referential, ( ) L' /X x t gh= , and are: a) the non-dimensional water elevation, Y/hL
( ), the non-dimensional elevation of the upper interface of the contact load layer h/hL + hc/hL ( ) and the non-dimensional bed elevation, Yb/hL ( ); b) the
non-dimensional flow velocity, L'U u gh= ( ), the non-dimensional unit
discharge, 3L'Q q gh= ( ) and the flow-averaged sediment concentration, C’
( ). Initial conditions and parameters of the closure equation as shown in table
4.1.
This result is in agreement with the experimental results shown in Leal et al. (2003). A more
complete discussion will be conducted latter. It should be noticed that the total discharge
increases across the shock associated to λ(2), which is not strange since the flux of
momentum is larger in constant state (2).
Solutions of type B are similar to those presented by Fraccarollo & Capart (2002). Figure
4.39 shows the structure of the solution of type B while the profiles of the primitive variables
are shown in figure 4.40. As is the case for type A solutions, the 1st characteristic field admits
377
a shock wave as a solution whereas the 3rd field admits an expansion wave. Contrarily to type
A, the solution for the λ(2)-field is an expansion wave (figures 4.37(b) and 4.39(b)).
The shock wave in the λ(1)-field is entropy satisfying, as it could be verified from figure
4.39(a) where it is visible that the characteristic lines converge into the shock path. Across
the rarefaction wave associated to the λ(3)– field, it is visible that the values of λ(1)
increase.
Characteristics path
5
4
3
2
1
0-2 -1 0 1 2 3 4
x (m)
t (s)
-2 -1 0 1 2 3 4
5
4
3
2
1
0
x (m)
t (s)
-2 -1 0 1 2 3 4
5
4
3
2
1
0
t (s)
FIGURE 4.39. Characteristic lines and Riemann wave structure for solution of type B. (a) λ(1)-
field; (b) λ(2)-field; and (c) λ(3)-field. Initial conditions and parameters of the closure
equation as shown in table 4.1. Adapted from Ferreira & Leal (1998).
(m)x
378
The characteristic lines bend as it was the case for the solution of type A. Across the
rarefaction wave associated to the λ(2)– field (figure 4.39b) the values of λ(1)
increase, as a
result of the increase of the flow velocity and the decrease of the water depth. Here resides
the main difference between solutions of types A and B in of what concerns the behaviour of
the 1st characteristic field.
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5X ' (-)
Z' (
-)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-1.5 -1 -0.5 0 0.5 1 1.5X ' (-)
U' ,
Q' ,
C'
(-)
FIGURE 4.40. Example of solution of type B. The profiles are shown in a self-similar
referential, ( ) L' /X x t gh= , and are: a) the non-dimensional water elevation, Y/hL
( ), the non-dimensional elevation of the upper interface of the contact load layer h/hL + hc/hL ( ) and the non-dimensional bed elevation, Yb/hL ( ); b) the
non-dimensional flow velocity, L'U u gh= ( ), the non-dimensional unit
discharge, 3L'Q q gh= ( ) and the flow-averaged sediment concentration, C’
( ). Initial conditions and parameters of the closure equation as shown in table
4.1.
As shown by Fracarollo & Armanini (1999) for the horizontal bed case, as the sediment
transport rate vanishes, the expansion waves of the 3rd and 2nd characteristic fields merge,
giving rise to the expansion wave of the Stoker solution.
379
4.5 CONCLUSIONS
A mathematical analysis of the systems of governing equations deduced in Chapters 2 and 3
was performed in this chapter. The emphasis was placed on the systems that express quasi-
equilibrium (or capacity) sediment transport, in which case the mass and momentum fluxes
among layers are zero. Mathematically, the absence of vertical fluxes means that, except for
the friction slope in the total momentum conservation equation, the systems are
homogeneous. Since the fundamental mathematical properties of a system of equations are
imposed by its homogeneous part (see §4.2.1.1, p. 306), it was decided to study the quasi-
equilibrium approximations only.
The applications of the model developed in Chapter 2 are not susceptible to feature
discontinuous flows, since the model was designed for low Froude numbers (see §2.1, p. 14).
On the contrary, the geomorphic shallow-water model developed in Chapter 3 was meant to
be applied in highly unsteady flow problems, where discontinuities, hydraulic jumps or bores,
are likely to be in the solution. Hence, most of the analysis is performed on the geomorphic
shallow-water equations, written as in §4.2.2.1.
Before analysing the full set of equations, some fundamental results on hyperbolicity, wave
propagation, ill-posedness and prescription of information at the computational boundaries is
reviewed. It is underlined that the computation of the direction of lines of constant phase,
associated to a given wave form, is amenable to an eigenvalue problem, expressed in the
characteristic polynomial (4.18) (see §4.2.1.3 and §4.2.1.4). The roots of that the
characteristic polynomial (the characteristics of the system) are the directions of the lines of
constant phase and express propagation of information in the space-time domain. Hence, a
well-posed hyperbolic problem possesses as many characteristic lines as dependent
variables (§4.2.1.4). It was also seen that, in the computational boundaries, this principle
applies and, as a consequence, independent information must be prescribed for each
characteristic line that “enters” the computational domain (§4.2.1.5 and §4.2.1.6).
Compatibility equations are also derived, as they are easily discretized.
Shock formation and weak solutions are discussed in §4.2.1.9 and §4.2.1.10. The conditions
for obtaining entropy admissible solutions are reviewed and the Rankine-Hugoniot equations
are derived for a general system of hyperbolic equations.
The governing conservation and closure equations, object of the analysis, are shown in
§4.2.2. Additional conditions for obtaining a solution are identified, addressing mostly the
problem of the growth of the contact-load layer. Indeed, if the thickness of the contact load
layer is a monotone increasing function of the mean flow velocity, a condition must be
imposed limiting that thickness to the local flow depth. The mathematical consequences,
namely on the continuity and the differentiability of the fluxes, are discussed.
Since discontinuities are expected, the system of conservation equations must be written in
conservative form. It is claimed in §4.2.3 that a strict conservative formulation is impossible
because the term that expresses the force of gravity in the momentum equation,
( ) ( ), ;b x bf h u Y⋅ ∂ , is, physically, not a flux and, mathematically, cannot be written in
conservative form. However, it is proved that, under the conditions of proposition 3.1, there
is indeed, a vector or conservative variables and, hence, solutions are possible.
380
The eingenstructure of the system of conservation equations is presented in §4.2.4. The
properties of the characteristic fields are discussed, namely signal and monotonicity. It is
shown in §4.2.5 that all fields are genuinely non-linear, which excludes the possibility of
featuring contact discontinuities as solutions.
The solutions of the Riemann problem for the geomorphic shallow water equations, which, in
some conditions can be used to describe the dam-beak problem (see §4.1), are identified in
§4.3. It is shown that solutions exist but are unique only in relation to the specific linearization
performed to the term ( ), ;bf h u ⋅ in ( ) ( ), ;b x bf h u Y⋅ ∂ . This is a consequence of the non-
purely conservative form of geomorphic shallow water equations.
Two types of solutions are encountered in §4.3.2. Type B solution is similar to that found by
Fraccarollo & Capart (2002) and it is composed of two rarefaction waves and a downstream
propagating shock. Type A solution features two shock waves, a faster bore-like shock and a
slower hydraulic jump-like discontinuity. It is shown that composite shock-rarefaction wave
solutions are not possible for the 3rd characteristic filed. The threshold conditions for the
occurrence of each of the solutions are investigated. In §4.3.3, the sets of equations
necessary to compute each of the solutions are presented and discussed.
Examples of computations of solutions of types A and B are shown in §4.4. Some properties
of the solution of type A are briefly discussed. A thorough analysis of the properties of the
Riemann solution is left to be performed in Chapter 5. It is envisaged that these theoretical
solutions may substitute the classic Stoker solution as the preferred reference situation for
geomorphic flows.
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HIRSCH, M., SMALE, S. & DEVANEY, R. L. (2004) Differential equations, dynamical systems & an
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384
385
ANNEX 4.1 – Calculation of the wave number of a wave form.
Consider the wave form (4.3), build as superposition of harmonic waves, weighted in phase
and amplitude, by ( )V k , where k is the wave number
( ( ) )( , ) ( ) dei tt
+∞+ω
−∞
= ∫ k x kV x V k ki (A4.1.1) ≡ (4.3)
It is shown that (A4.1.1) can be transformed into (4.4). Formally, (4.4) is advantageous for the
manipulations performed in §4.2.
Let the wave number be expressed by an asymptotic expansion around κ, truncated to first
order
= + εk κ η (A4.1.2)
The wave number κ should be chosen as that of the most representative harmonic or, simply
as the average wave number
( ) dp+∞
−∞
= ∫κ k k k (A4.1.3)
where ( )p κ is an appropriate function of density of probability. For the sake of simplicity it
can be chosen as that of the uniform distribution. I any case, the chosen κ must allow for the
expansion of the angular frequency in a Taylor series around ( )ω ≡ ϖκ . The angular
frequency becomes
( )3T12( ) ( ) Oω = ω + ∇ω + +κ κk κ k k k ki H (A4.1.4)
where H is the Hessian matrix of the angular frequency (the matrix of second derivatives).
The case of non-dispersive waves is of particular interest in this text. In this case =H 0 .
Introducing (A4.1.2) and (A4.1.4) in (A4.1.1) and writing 2( ) ( )+ ε = εV κ η f η , the following
manipulations are possible
i( ) i( ( ) ( ) ) 2
i( ) i( ( ) ( ) ) 22
i( ) i ( ) i ( ( ) )
( , ) ( ) d
( ) d
( ) d
e e
e e
e e e
t t t
t t t
t t t
t+∞
+ϖ ε +∇ ω +ε∇ ω
−∞+∞
+ϖ ε +∇ ω +ε∇ ω
−∞+∞
+ϖ ∇ ω ε +∇ ω
−∞
= + ε ε =
ε =ε
=
∫∫
∫
κ x η x κ η
κ x η x κ η
κ x κ η x
V x V κ η η
f η η
f η η
i i i i
i i i i
i i i
386
i( ) i i
i( )0
( ) d
( , )
e e e
e
t
t
+∞+ϖ − τ
−∞+ϖ
=
τ
∫κ x η χ
κ x
f η η
V χ
i i
i
(A4.1.5)
In (A4.1.5) a change of variables is performed. The following identities are used
( ) tτ = −∇ ω κi (A4.1.6)
( )( ) t= ε − ∇ ωχ x (A4.1.7)
The new phase of the wave form is, after these manipulations, written as
( , )t tΣ = − ϖx κ xi (A4.1.8)
where ( )ϖ ≡ ω κ . Formally, the wave can thus be written as
i ( , )
0( , ) ( , )e tt t Σ= xV x V x (A4.1.9) ≡ (4.4)
The specification of 0 ( , )tV x requires initial conditions and this function is sought sufficiently
regular to admit Fourier transforms. From (A4.1.9) it is clear that the propagation of the wave
form is formally the propagation of 0 ( , )tV x along the surfaces of constant phase, the latter
expressed by the exponential term.
387
ANNEX 4.2 – Coefficients of the matrixes A and B of equation (4.91)
The coefficients of the matrixes A and B are presented. These read
2 3
1 2
1 0 10
1K K
N N p
= ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦
A 4 5 6
4 5
0
0
u hK K KN N
= ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
B (A4.2.1)(a)(b)
It should be noticed that matrix A as expressed by equation (4.91) was subjected to a
pivoting operation in order to eliminate the coefficient of ( )t h∂ in the momentum equation.
Obviously, the operation implies not only the indroduction of K3 but also of a redefinition of
the second line of B. Traditionally, the full non-conservative momentum equation has been
written with a term proportional to ( )t bY∂ . In fluvial regimes, i.e., for low Froude numbers,
this term is neglected on physical basis. In a dam-break flow, ( )t bY∂ may be of the order of
( )t u∂ and either of the formulations can be retained. For the purpose of comparison with
previous works, the term proportional to ( )t bY∂ was preferred.
The coefficients are
( )( )( )2 1 ( 1) uK h s C C u= + − + ∂ (A4.2.2)
( )( )( )3 1 ( 1) hK u s C C h= − + − + ∂ (A4.2.3)
1 2 3 4 5 6
4 4 4 4 4 4 4K K K K K K K= + + + + + (A4.2.4)
where
( ) ( ) ( ) ( ) ( )( )14 ( 1) 2 1 ( 1) 1 ( 1)c h c c c c h c c c h c cK u s C h u s C u h s C h u= − ∂ + + − ∂ + + − ∂
( ) ( ) ( ) ( ) ( )( )24 ( 1) 2 1 ( 1) 1 ( 1)s h s s s s h s s s h s sK u s C h u s C u h s C h u= − ∂ + + − ∂ + + − ∂
( ) ( ) ( )( )3 2'4 2 ( 1) 2 1 ( 1)k
h c c c h c cK g s C h s C h h= − ∂ + + − ∂
( ) ( ) ( )( )4 214 2 ( 1) 2 1 ( 1)h s s s h s sK g s C h s C h h= − ∂ + + − ∂
( ) ( ) ( ) ( ) ( )( )54 ' ( 1) 1 ( 1) 1 ( 1)h s s c s h s c s h c sK k g s C h h s C h h s C h h= − ∂ + + − ∂ + + − ∂
( )( )( )6 24 1 ( 1) hK u s C C h= − + − + ∂
1 2 3 4 5 6
5 5 5 5 5 5 5K K K K K K K= + + + + + (A4.2.5)
where
( ) ( ) ( ) ( ) ( )( )15 ( 1) 2 1 ( 1) 1 ( 1)c u c c c c u c c c u c cK u s C h u s C u h s C h u= − ∂ + + − ∂ + + − ∂
388
( ) ( ) ( ) ( ) ( )( )25 ( 1) 2 1 ( 1) 1 ( 1)s u s s s s u s s s u s sK u s C h u s C u h s C h u= − ∂ + + − ∂ + + − ∂
( ) ( ) ( )( )3 2'5 2 ( 1) 2 1 ( 1)k
u c c c u c cK g s C h s C h h= − ∂ + + − ∂
( ) ( ) ( )( )4 215 2 ( 1) 2 1 ( 1)u s s s u s sK g s C h s C h h= − ∂ + + − ∂
( ) ( ) ( ) ( ) ( )( )55 ' ( 1) 1 ( 1) 1 ( 1)u s s c s u s c s u c sK k g s C h h s C h h s C h h= − ∂ + + − ∂ + + − ∂
( )( )( )65 1 ( 1) hK uh s C C h= − + − + ∂
6K ( )( )1 ( 1) ( 1)c c s c cg s C h h gh g s C h= + − + = + − (A4.2.6)
( ) ( )( )1 h c c h s sN C h C h= ∂ + ∂ (A4.2.7)
( ) ( )( )2 u c c u s sN C h C h= ∂ + ∂ (A4.2.8)
( )( )4 hN u C C h= + ∂ (A4.2.9)
( )( )5 uN h C C u= + ∂ (A4.2.10)
The coefficients of the characteristic polynomial, equation (4.94), are
[ ]1 6 2 5 2 5 1 4 2 2 41 (1 ) (1 )a K N K p K p u K N K N K N
Dλ= − − − − + − +
( )3 5 2 11 K N N u N h
Dλ⎡ ⎤+ + −⎣ ⎦ (A4.2.11)
( )2 6 1 4 5 6 5 21 (1 ) (1 )a hK N hK p K p u K N N u
Dλ⎡ ⎤= − − + − − +⎣ ⎦
( ) ( )4 5 4 5 3 4 3 51 K N N K hK N K N u
Dλ⎡ ⎤+ − + −⎣ ⎦ (A4.2.12)
( )3 6 5 41a K uN hN
Dλ= − (A4.2.13)
where
2 2 1 3 2(1 )D K p K N K Nλ = − − − (A4.2.14)