profound things

AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

AA214B: NUMERICAL METHODS FORCOMPRESSIBLE FLOWS

The Finite Difference Method

These slides are based on the recommended textbook: Culbert B. Laney. ComputationalGas Dynamics, CAMBRIDGE UNIVERSITY PRESS, ISBN 0-521-62558-0



Outline

1 Conservative Finite Difference Methods in One Dimension

2 Forward, Backward, and Central Time Methods

3 Domain of Dependence and CFL Condition

4 Linear Stability Analysis

5 Formal, Global, and Local Order of Accuracy

6 Upwind Schemes in One Dimension

7 Nonlinear Stability Analysis

8 Multidimensional Extensions



Note: The material covered in this chapter equally applies to scalarconservation laws and to the Euler equations, in one and multipledimensions. In order to keep things as simple as possible, it is presentedin most cases for scalar conservation laws: first in one dimension, then inmultiple dimensions.



Conservative Finite Difference Methods in One Dimension

Recall that scalar conservation laws are simple scalar models of the Euler equations that canbe written in strong conservation form as

u

t+f (u)

x= 0 (1)

Suppose that a 1D space is divided into grid points xi and cells [xi1/2, xi+1/2], wherexi+1/2 is called a cell edge

Also suppose that time is divided into time-intervals [tn, tn+1]

The conservation form of a finite difference method applied to the numerical solution ofequation (1) is defined as follows

t

(u

t

)ni

= (f ni+1/2 f ni1/2) (2)

where the subscript i designates the point xi , the superscript n designates the time tn, a

hat designates a time-approximation, and

=t

x, t = tn+1 tn, x = xi+1/2 xi1/2




One interpretation of the finite difference approach (2) and theconservation form label is the approximation of the following integralform of equation (1)

1

x

xi+1/2xi1/2

[u(x , tn+1) dx u(x , tn)] dx

= 1x

tn+1tn

[f(u(xi+1/2, t)

) f (u(xi1/2, t))] dtwhich clearly describes a conservation law




Not every finite difference method can be written in conservationform: Those which can are called conservative and their associatedquantities f ni+1/2 are called conservative numerical fluxes

finite difference methods derived from the conservation form of theEuler equations or scalar conservation laws tend to be conservativefinite difference methods derived from other differential forms (forexample, primitive or characteristic forms) of the aforementionedequations tend not to be conservativeconservative finite differencing implies correct shock and contactlocations




Like many approximation methods, conservative finite differencemethods can be divided into implicit and explicit methods

in a typical implicit method(u

t

)ni

=

(u

t

)(uniK1 , ..., u

ni+K2 ; u

n+1iL1 , ..., u

n+1i+L2

)

f ni+1/2 = f (uniK1+1, ..., u

ni+K2 ; u

n+1iL1+1, ..., u

n+1i+L2

) (3)

so that from (2) one has

un+1i = u(uniK1 , ..., u

ni+K2 ; u

n+1iL1 , ..., u

n+1i , ..., u

n+1i+L2

) (4)

= the solution of a system of equations is required at eachtime-step

Note: if uniK1+1 in (3) were written as uniK1 , one would get the less

convenient notation un+1i = u(uniK11, ..., u

ni+K2

; un+1iL1 , ..., un+1i+L2

)instead of (4)




Like many approximation methods, conservative finite differencemethods can be divided into implicit and explicit methods(continue)

in a typical explicit method(u

t

)ni

=

(u

t

)(uniK1 , ..., u

ni+K2 ; u

n+1i )

f ni+1/2 = f (uniK1+1, ..., u

ni+K2 )

so that from (2) one has

un+1i = u(uniK1 , ..., u

ni+K2 )

= only function evaluations are incurred at each time-step(uniK1 , ..., u

ni+K2

) and (un+1iL1 , ..., un+1i+L2

) are called the stencil or direct

numerical domain of dependence of un+1iK1 + K2 + 1 and L1 + L2 + 1 are called the stencil widths




Summary: typical stencil diagram




Like any proper numerical approximation, proper finite differenceapproximation becomes perfect in the limit x 0 and t 0

an approximate equation is said to be consistent if it equals the trueequations in the limit x 0 and t 0a solution to an approximate equation is said to be convergent if itequals the true solution of the true equation in the limit x 0 andt 0

Hence, a conservative approximation is consistent when

f (u, ..., u) = f (u)

= in this case, the conservative numerical flux f is said to beconsistent with the physical flux

A conservative numerical method and therefore a conservativefinite difference method automatically locates shocks correctly(however, it does not necessarily reproduce the shape of the shockcorrectly)

A method that explicitly enforces the Rankine-Hugoniot relation iscalled a shock-capturing method



Forward, Backward, and Central Time Methods

Forward Time Methods

Forward Time (FT) conservative finite difference methodscorrespond to the choices

t

(u

t

)ni

= un+1i uni and f ni+1/2 = f (uniK1+1, ..., uni+K2 )

with Forward Space (FS) approximation of the termu

x(xi , t

n), this

leads to the FTFS scheme

un+1i = uni (f ni+1/2 f ni1/2), with f ni+1/2 = f (uni+1)

with Backward Space (BS) approximation of the termu

x(xi , t

n),

this leads to the FTBS scheme

un+1i = uni (f ni+1/2 f ni1/2), with f ni+1/2 = f (uni )

with Central Space (CS) approximation of the termu

x(xi , t

n), this

leads to the FTCS scheme

un+1i = uni (f ni+1/2 f ni1/2), with f ni+1/2 = 12 (f (u

ni+1) + f (u

ni ))




Backward Time Methods

Backward Time (BT) conservative finite difference methodscorrespond to the choices

t

(u

t

)ni

= un+1i uni and f ni+1/2 = f (un+1iK1+1, ..., un+1i+K2 )


x(xi , t

n), this

leads to the BTFS scheme

un+1i = uni (f ni+1/2 f ni1/2), with f ni+1/2 = f (un+1i+1 )


x(xi , t

n),

this leads to the BTBS scheme

un+1i = uni (f ni+1/2 f ni1/2), with f ni+1/2 = f (un+1i )


x(xi , t

n), this

leads to the BTCS scheme

un+1i = uni (f ni+1/2 f ni1/2), with f ni+1/2 = 12

(f (un+1i+1 ) + f (u

n+1i )

)AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS



Central Time Methods

Central Time (CT) conservative finite difference methods correspondto the choices

t

(u

t

)ni

=1

2(un+1i un1i ) and f ni+1/2 = f (uniK1+1, ..., uni+K2 )


x(xi , t

n), this

leads to the CTFS scheme

un+1i = un1i 2(f ni+1/2 f ni1/2), with f ni+1/2 = f (uni+1)


x(xi , t

n),

this leads to the CTBS scheme

un+1i = un1i 2(f ni+1/2 f ni1/2), with f ni+1/2 = f (uni )


x(xi , t

n), this

leads to the CTCS scheme

un+1i = un1i 2(f ni+1/2 f ni1/2), with f ni+1/2 =

1

2(f (uni+1) + f (u

ni ))



Domain of Dependence and CFL Condition

Numerical and Physical Domains of Dependence

Recall the theory of characteristics: A point in the x t plane isinfluenced only by points in a finite domain of dependence andinfluences only points in a finite range of influence

Hence, the physical domain of dependence and physical range ofinfluence are bounded on the right and left by the waves with thehighest and lowest speeds

In a well-posed problem, the range of influence of the initial andboundary conditions should exactly encompass the entire flow in thex t plane





The direct numerical domain of dependence of a finite differencemethod is its stencil: For example, if the solution approximated byan implicit finite difference method can be written as

un+1i = u(uniK1 , ..., u

ni+K2 ; u

n+1iL1 , ..., u

n+1i+L2

)

its direct numerical domain of dependence is the region of the x tplane covered by the points (uniK1 , ..., u

ni+K2

; un+1iL1 , ..., un+1i+L2

)Similarly, if the solution approximated by an explicit finite differencemethod can be written as

un+1i = u(uniK1 , ..., u

ni+K2 )

its direct numerical domain of dependence is the region of the x tplane covered by the points (uniK1 , ..., u

ni+K2

)The full (or complete) numerical domain of dependence of a finitedifference method consists of the union of its direct numericaldomain of dependence and the domain covered by the points of thex t plane upon which the numerical values in the direct numericaldomain of dependence depend upon





The Courant-Friedrichs-Lewy or (CFL) conditionThe full numerical domain of dependence must contain the physicaldomain of dependence

Any numerical method that violates the CFL condition missesinformation affecting the exact solution and may blow up to infinity:For this reason, the CFL condition is necessary but not sufficient fornumerical stability




Scalar Conservation Laws

Consider first the linear advection problem

u

t+ a

u

x= 0

u(x , 0) =

{1 if x < 00 if x 0

Assume that a > 0: The exact solution is

u(x , t) = u(x at, 0) ={

1 if x at < 00 if x at 0

The FTFS approximation with x = cst is

un+1i = (1 + a)uni auni+1

u0i = u(ix , 0) =

{1 if i < 00 if i 0

where as before, = tx





Then

u1i =

1 if i 21 + a if i = 10 if i 0

u2i =

1 if i 3

(1 + a)(1 a) if i = 2(1 + a)(1 + a) if i = 1

0 if i 0and so forth

The first two time-steps reveal that FTFS moves the jump in thewrong direction (left rather than right!) and produces spuriousoscillations and overshoots

Furthermore, the exact solution yields u(0,t) = 1, but FTFS yieldsu10 = 0!





This is because FTFS violates the CFL condition

un+1i = (1 + a)uni auni+1





FTCS satisfies the CFL condition

However, it almost always blow up (as will be seen in a homework):This illustrates the fact that the CFL condition is a necessary butnot sufficient condition for numerical stability

You can also check that when applied to the solution of any scalarconservation law, the BTCS method always satifies the CFLcondition





For scalar conservation laws, the CFL condition translates into asimple inequality restricting the wave speed

linear advection equation and explicit forward-time method withun+1i = u(u

niK1 , ..., u

ni+K2

)in the x t plane, the physical domain of dependence is the line of slope 1/ain the x t plane, the full numerical domain of dependence of un+1i is boundedon the left by a line of slope

t

K1x=

K1and on the right by a line of slope

tK2x

= K2

hence, the CFL condition is

K2 a K1

K2 a K1

which requires that waves travel no more than K1 points to the right or K2points to the left during a single time-step





For scalar conservation laws, the CFL condition translates into asimple inequality restricting the wave speed (continue)

linear advection equation and explicit forward-time method withun+1i = u(u

niK1 , ..., u

ni+K2

) (continue)

if K1 = K2 = K , the previous CFL condition becomes

|a| Kfor this reason, a is called the CFL number or the Courant numberkeep in mind however that in general, a = a(u) and therefore the CFL numberdepends in general on the solutions range





For scalar conservation laws, the CFL condition translates into asimple inequality restricting the wave speed (continue)

linear advection equation and implicit backward-time method withun+1i = u(u

niK1 , ..., u

ni+K2

; un+1iL1 , ..., un+1i+L2

)

if L1 > 0 and L2 = 0, the full numertical domain of dependence ofun+1i includes everything to the left of x = xi and beneath t = t

n+1

in the x t planeif L1 = 0 and L2 > 0, the full numertical domain of dependence ofun+1i includes everything to the right of x = xi and beneath t = t

n+1

in the x t planeif L1 > 0 and L2 > 0, the full numertical domain of dependence ofun+1i includes everything in the entire x t plane beneath t = tn+1conclusion: as long as their stencil includes one point to the left andone to the right, implicit methods avoid CFL restrictions by using theentire computational domain (hence, this includes BTCS but notBTFS or BTBS)




The Euler Equations

In 1D, the Euler equations have three families of waves that definethe physical domain of dependence

For each family of waves, a CFL condition of a given numericalmethod can be established as in the case of a scalar conservationlaw: Then, the overall CFL condition is the most restrictive of allestablished CFL conditions

For example, if K1 = K2 = K , A is the Jacobian matrix of theconservative flux vector, and (A) denotes its spectral radius((A) = max (|vx a|, |vx |, |vx + a|

), the CFL condition of an

explicit forward-time method becomes(recall (22)

)(A) K

(A) is called the CFL number or the Courant number




The Euler Equations

(A) K

For supersonic flows, all waves travel in the same direction, eitherleft or right the minimum stencil allowed by the CFL conditioncontains either W ni1 and W

ni for right-running supersonic flow, or

W ni and Wni+1 for left-running supersonic flow

For subsonic flows, waves travel in both directions, and theminimum stencil should always contain W ni1, W

ni , and W

ni+1

Hence, a smart or adaptive stencil can be useful for the case of theEuler equations!



Linear Stability Analysis

Unstable solutions exhibit significant spurious oscillations and/orovershoots

Unstable solutions of linear problems exhibit unbounded spuriousoscillations: Their errors grow to infinity as t Hence the concept of instability discussed here for the solution oflinear problems is that of ubounded growth

Since unstable solutions typically oscillate, it makes sense to describethe solution of a linear problem such as a linear advection problemas a Fourier series (sum of oscillatory trigonometric functions)




von Neumann Analysis

The Fourier series for the continuous (in space) solution u(x , tn) onany spatial domain [a, b] is

u(x , tn) = an0 +m=1

anm cos

(2pim

x ab a

)+m=1

bnm sin

(2pim

x ab a

)(5)

For the discrete solution uni u(xi , tn), the Fourier series is obtainedby sampling (5) as follows

uni = an0 +

m=1

anm cos

(2pim

xi ab a

)+m=1

bnm sin

(2pim

xi ab a

)(6)

Assume xi+1 xi = x = cst, x0 = a, and xN = b xi a = ixand b a = Nx : This transforms (6) into

uni = an0 +

m=1

(anm cos

(2pim

i

N

)+ bnm sin

(2pim

i

N

))(7)





Recall that samples can only support wavelengths of 2x or longer(the Nyquist sampling theorem states that samples spaced apart byx perfectly represent functions whose shortest wavelengths are4x): Hence (7) is truncated as follows

uni an0 +N/2m=1

(anm cos

(2pimi

N

)+ bnm sin

(2pimi

N

))An equivalent expression in the complex plane using I as thenotation for the pure imaginary number (I 2 = 1) is

uni N/2

m=N/2C nme

I 2pimiN =

N/2m=N/2

unim (8)

From (7), (8), and Eulers formula e I = cos + I sin it follows that

C n0 = an0 , C

nm =

anm Ibnm2

, C nm =anm + Ib

nm

2





Hence, each term of the Fourier series can be written as

unim = Cnme

(I 2pimiN ) = C nmeIm i

where m =2pim

Nand m = N/2, ,N/2

Because of linearity the amplification factor

Gm =C n+1mC nm

= Gm()

does not depend on n: However, it depends on (since un+1i and uni

are produced by the numerical scheme being analyzed) which itselfdepends on t

Hence, each term of the Fourier series can be expressed as

unim =C nmC n1m

C2m

C 1m

C 1mC 0m

C 0meIm i = Gm GmC 0me Im i = G nmC 0me Im i

where G nm = Gnm() = (Gm())

n





Finally, assume that C 0m = 1 (for example): This leads to

unim = Gnm()e

Im i

Conclusions

the linear approximation is linearly stable if |Gm()| < 1 for all mit is neutrally linearly stable if |Gm()| 1 for all m and |Gm()| = 1for some mit is linearly unstable if |Gm()| > 1 for some m

Each of the above conclusion can be re-written in terms of =t

xApplication (in class): apply the von Neumann analysis to determinethe stability of the FTFS scheme for the linear advection equation




Matrix Method

Shortcomings of the von Neumann stability analysis method

requires the solution to be periodic (un0 = unN)

requires constant spacing xdoes not account for the boundary conditions

Alternative method: so-called Matrix (eigenvalue analysis) Methodbased on the fact that for a linear problem and a linearapproximation method, one has

un+1 = M()un, where un = (uno , un1 , , unN)T

and M is an amplification matrix which depends on theapproximation scheme and on

This impliesun = Mn()u0




Matrix Method

un = Mn()u0

Suppose that M is diagonalizable

M() = Q1()Q, = diag (1(), , N())

ThenMn() = Q1nQ (Qu)n = n()(Qu)0

Conclusions

the linear approximation is linearly stable if (M()) < 1it is neutrally linearly stable if (M()) = 1it is linearly unstable if (M()) > 1




Matrix Method

Advantages of the Matrix Method for (linear) stability analysisdoes not require the solution to be periodicdoes not require constant grid spacingincorporates the effects of the boundary conditions

Shortcoming: in general, the computation of (M) is not trivialHowever, the above shortcoming is not an issue when the objectiveis to prove the unconditional stability of an (implicit) scheme

re-write the linear version of equation (2) in matrix form as

du

dt+ B(x)u = 0 (9)

suppose that B is diagonalizable and transform equation (9) into the set ofindependent scalar equations

dvi

dt+ i (x)vi = 0, i = 1, , N

focus on one of the above equations and discretize it in timeapply the scalar form of the Matrix Method for stability analysis: if the conclusionturns out to be independent of i , then the aforementioned shortcoming is not anissueexample (in class): apply the Matrix Method to determine the stability of a BTscheme for the linear advection equation



Formal, Global, and Local Order of Accuracy

Formal order of accuracy measures the order of accuracy of theindividual space and time approximations

Taylor series expansionsmodified linear equations

However due to instability, formal order of accuracy may not beindicative of the actual performance of a method: For example,recall that a stability condition is (A) K t(A) Kxand observe that such a stability condition prevents, for example,fixing t and studying the order of accuracy of the individual spaceapproximation when x 0




Besides formal order of accuracy, one way to measure the order ofaccuracy is to reduce x and t simultaneously while maintaining

=t

xconstant and fixing the initial and boundary conditions

In this case, a method is said to have global p-th order of accuracy(in space and time) if

e Cxxp = Cttp, ei = u(xi , tni ) uni

for some constant Cx and the related constant Ct =Cxp

Other error measures can be obtained by using the 1-norm, 2-norm,or any vector norm, or if the error is measured pointwise




Determining analytically the global order of accuracy defined abovecan be challenging: For this reason, it is usually predicted bycomparing two different numerical solutions obtained using the samenumerical method but two different values of x

p =log(e2/e1)

log(x2/x1)=

log(e2/e1)log(t2/t1)

where eli = |u(xi , tn) uni | is the absolute error for xl andtl = xl , l = 1, 2




Another way to measure the order of accuracy is to assume that thesolution is perfect at time tn that is, uni = u(xi , t

n)i , which isusually true for n = 0 and measure the local (in time) truncationerror induced by a single time-step

ei =u(xi , t

n+1) un+1it

Now, let t 0 and x 0 while maintaining = tx

, and the

initial and boundary conditions fixed: Then, a method is said tohave local p-th order of accuracy (in space and time) if

ei Cxxp = Cttp

for some constant Cx and the related constant Ct =Cxp

Unlike the global order of accuracy, the local order of accuracy isrelatively easy to determine analyticallyExample (in class): determine analytically the local order ofaccuracy of the FTFS scheme for the linear advection equation



Upwind Schemes in One Dimension

In 1D, there are right-running waves and left-running waves: Forright-running waves, right is the downwind direction and left is theupwind direction

Similarly for left-running waves, left is the downwind direction andright is the upwind direction

Then every numerical approximation to a scalar conservation lawcan be described as

Centered: if its stencil contains equal numbers of points in bothdirectionsUpwind: if its stencil contains more points in the upwind directionDownwind: if its stencil contains more points in the downwinddirection

Upwind and downwind stencils are adjustable or adaptive stencils:Upwind and downwind methods test for wind direction and then,based on the outcome of the tests, select either a right- or aleft-biased stencil




Upwinding ensures shock avoidance if the shock reverses the wind,whereas central differencing does not




Upwinding does not ensure shock avoidance if the shock does notreverse the wind

downwinding on the right above avoids the shock but violates theCFL condition and thus would create larger errors than crossing theshock would




General remarks

upwind methods are popular because of their excellent shockcapturing abilityamong simple FT or BT methods, upwind methods outdo centeredmethods: However, higher-order upwind methods often have nospecial advantages over higher-order centered methods

Sample techniques for designing methods with upwind and adaptivestencils

flux averaging methodsflux splitting methods?

wave speed splitting methods?




Introduction to Flux Splitting

Flux splitting is defined as

f (u) = f +(u) + f (u)df +

du 0, df

du 0

Hence, f +(u) is associated with a right-running wave and f (u) isassociated with a left-running wave

Using flux splitting, the governing conservation law becomes

u

t+f +

x+f

x= 0

which is called the flux split form

Then,f +

xcan be discretized conservatively using at least one point

to the left, andf

xcan be discretized conservatively using at least

one point to the right, thus obtaining conservation and satisfactionof the CFL condition





Unfortunately, flux splitting cannot describe the true connectionbetween fluxes and waves unless all waves run in the same direction

if all waves are right-running, the unique physical flux splitting isf + = f and f = 0if all waves are left-running, the unique physical flux splitting isf = f and f + = 0

This is the case only for scalar conservation laws away from sonicpoints, and for the supersonic regime in the Euler equations





Assume thatf +

xis discretized with a leftward bias so that the

approximation at x = xi is centered or biased towards x = xi1/2(f +

x

)i

f +i1/2

xfor some f +i1/2

Assume thatf

xis discretized with a rightward bias so that the

approximation at x = xi is centered or biased towards x = xi+1/2(f

x

)i

f i+1/2

xfor some f i+1/2

Using forward Euler to perform the time-discretization leads to

un+1i = uni (f +

n

i1/2 + fni+1/2)

which is called the flux split form of the numerical approximation





A method in flux split form is conservative if and only if

f +n

i+1/2 + fni+1/2 = g

ni+1 gni for some gni (10)

Proofun+1i = u

ni (f +

n

i1/2 + gni gni + f

n

i+1/2)

compare with the conservation form un+1i = uni (f ni+1/2 f ni1/2)

= f ni+1/2 = f n

i+1/2 + gni , f

ni1/2 = f +

n

i1/2 + gni

= f ni+1/2 = f ni+1/2 gni , f +n

i1/2 = f ni1/2 + gnirequire now thatf ni+1/2 = f

n(i+1)1/2 f

n

i+1/2 + gni = f +

n

i+1/2 + gni+1

= f +ni+1/2 + f n

i+1/2 = gni+1 gni

Since there are no restrictions on gni , every conservative method hasinfinitely many flux split forms that are useful for nonlinear stabilityanalysis





Example: Design a first-order upwind method for Burgers equationusing flux splitting then re-write it in conservation form

for Burgers equation, the unique physical flux splitting is

f (u) =u2

2= max(0, u)

u

2 f +(u)

+ min(0, u)u

2 f(u)





Example: Design a first-order upwind method for Burgers equationusing flux splitting then re-write it in conservation form (continue)

a flux split form of Burgers equation is

u

t+

1

2

x(max(0, u)u) +

1

2

x(min(0, u)u) = 0

a backward-space approximation off +

xgives(

x(max(0, u)u)

)ni

max(0, uni )u

ni max(0, uni1)uni1

x

a forward-space approximation off

xgives(

x(min(0, u)u)

)ni

min(0, uni+1)u

ni+1 min(0, uni )unix

combining these with a FT approximation yields

un+1i = uni 2 (max(0, u

ni )u

ni max(0, uni1)uni1)

2

(min(0, uni+1)uni+1 min(0, uni )uni )





Example: Design a first-order upwind method for Burgers equationusing flux splitting then re-write it in conservation form (continue)

the reader can check that the first-order upwind method described inthe previous page can be re-written in conservation form using

gni =1

2(uni )

2




Introduction to Wave Speed Splitting

In contrast with flux splitting, wave speed splitting uses thegoverning equations in non conservation form and tends to yield nonconservative approximations

Hence in most cases, flux splitting is preferred over wave speedsplitting ... except when the flux function has the property

f (u) =df

duu = a(u)u

which means that f (u) is a homogeneous function of degree 1: Thisproperty makes flux splitting and wave speed splitting closely related





For scalar conservation laws, wave speed splitting can be written as

a(u) = a+(u) + a(u)a+(u) 0, a(u) 0

Then, the scalar conservation law can be written as

u

t+ a+

u

x+ a

u

x= 0

which is called the wave speed split form

Then, a+u

xcan be discretized conservatively using at least one

point to the left, and au

xcan be discretized conservatively using

at least one point to the right, thus obtaining satisfaction of theCFL condition





Next, consider vector conservation laws such as the Euler equations

Split the Jacobian matrix as follows

A(u) = A+(u) + A(u)

where the eigenvalues of A+ are non negative and those of A arenon positive

A+ 0, A 0Recall that A+ and A are obtained by computing and splitting theeigenvalues of A

The wave speed split form of the Euler equations can then bewritten as

u

t+ A+

u

x+ A

u

x= 0

Again, A+u

xcan then be discretized conservatively using at least

one point to the left, and Au

xusing at least one point to the

right, thus obtaining satisfaction of the CFL condition





If f is a homogeneous function of degree 1, then

f = Au f = Au

However, the above flux vector splitting may or may not satisfydf +

du 0 and df

du 0





Assume that a+u

xis discretized with a leftward bias so that the

approximation at x = xi is centered or biased towards x = xi1/2(a+u

x

)ni

a+ni1/2uni uni1

x

Assume that au

xis discretized with a rightward bias so that the

approximation at x = xi is centered or biased towards x = xi+1/2(au

x

)ni

ani+1/2uni+1 uni

x

Using forward Euler to perform the time-discretization leads to

un+1i = uni a

n

i+1/2(uni+1 uni ) a+

n

i1/2(uni uni1)

which is called the wave speed split form of the numericalapproximation





The flux split form and wave speed form are connected via

f n

i+1/2 = ani+1/2(u

ni+1 uni )

From the above relation and equation (10), it follows that

(a+n

i+1/2 + ani+1/2)(u

ni+1 uni ) = gni+1 gni for some flux function gni

(11)

Hence, the transformation from conservation form to wave speedform and vice versa is

f ni+1/2 = ani+1/2(u

ni+1 uni ) + gni , f ni1/2 = a+

n

i1/2(uni uni1) + gni

(12)





un+1i = uni a

n

i+1/2(uni+1 uni ) a+

n

i1/2(uni uni1)

The above notation for the wave speed split form is the standardnotation when wave speed splitting is used to derive newapproximation methods

Wave speed split form is also often used as a preliminary step innonlinear stability analysis, in which case the standard notation is

un+1i = uni + C

+n

i+1/2(uni+1 uni ) C

n

i1/2(uni uni1)

Hence

C+n

i+1/2 = an

i+1/2, Cni1/2 = a

+n

i1/2 Cn

i+1/2 = a+n

i+1/2

(13)





From (13), it follows that if a method is derived using wave speedsplitting and not just written in wave speed split form, the splittingunderlying (11) can also be written as

a(u) = C(u) C+(u), C+(u) 0,C(u) 0

Then, the conservation condition (11) becomes

(Cn

i+1/2 C+n

i+1/2)(uni+1 uni ) = (gni+1 gni )

And equations (12) become

f ni+1/2 = C+n

i+1/2(uni+1uni )+gni , f ni1/2 = C

n

i1/2(uni uni1)+gni



Nonlinear Stability Analysis

Focus is set here on explicit FT difference approximations

Recall that unstable solutions exhibit significant spurious oscillationsand/or overshoots

Recall also that linear stability analysis focuses on these oscillationsand relies on the Fourier series representation of the numericalsolution: It requires only that this solution should not blow up, ormore specifically, that each component in its Fourier seriesrepresentation should not increase to infinity

because of linearity, this is equivalent to requiring that eachcomponent in the Fourier series should shrink by the same amount orstay constant at each time-step




Similarly, nonlinear stability analysis focuses on the spuriousoscillations of the numerical solution, but without representing it bya Fourier series

it can require that the overall amount of oscillation remains bounded,which is known as the Total Variation Bounded (TVB) conditionit can also require that the overall amount of oscillation, as measuredby the total variation, either shrinks or remains constant at eachtime-step

(this is known as the Total Variation Diminishing (TVD)

condition)

however, whereas not blowing up and shrinking are equivalentnotions for linear equations, these are different notions for nonlinearequations: In particular, TVD implies TVB but TVB does notnecessarily imply TVD




Monotonicity Preservation

The solution of a scalar conservation law on an infinite spatialdomain is monotonicity preserving: If the initial condition ismonotone increasing (decreasing), the solution is monotoneincreasing (decreasing) at all times

Suppose that a numerical approximation inherits this monotonicitypreservation property: Then, if the initial condition is monotone, thenumerical solution cannot exhibit a spurious oscillationMonotonocity preservation was first suggested by the Russianscientist Godunov in 1959: It is a nonlinear stability condition, butnot a great one for the following reasons:

it does not address the case of nonmonotone solutionsit is a too strong condition:

it does not allow even an insignificant spurious oscillation that doesnot threaten numerical stabilityattempting to purge all oscillatory errors, even the small ones, maycause much larger nonoscillatory errors

Godunovs theorem: For linear methods (NOT to be confused withlinear problems), monotonicity preservation leads to first-orderaccuracy at best




Total Variation Diminishing

TVD was first proposed by the american applied mathematicianAmiram Harten in 1983 as a nonlinear stability condition

The total variation of the exact solution may be defined as follows

TV (u(, t)) = supi

i=

|u(xi+1, t) u(xi , t)|

Laney and Caughey (1991):

the total variation of a function on an infinite domain is a sum ofextrema maxima counted positively and minima countednegatively with the two infinite boundaries always treated asextrema and counting each once, and every other extrema countingtwice





Numerical effects that can cause the total variation to increase





The exact solution of a scalar conservation law is TVD

TV (u(, t2)) TV (u(, t1)) , t2 t1What about the numerical solution of a scalar conservation law?

The total variation of a numerical approximation at time tn may beequally defined as

TV (un) =

i=|uni+1 uni |

it is the sum of extrema maxima counted positively and minimacounted negatively with the two infinite boundaries always treatedas extrema and counting each once, and every other extremacounting twice

Now, a numerical approximation inherits the TVD property if

n, TV (un+1) TV (un)AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS




Important result: TVD implies monotonicity preservation andtherefore implies nonlinear stability

Proof: Suppose that the initial condition is monotone

the TV of the initial condition is u u if it is monotoneincreasing and u u if it is monotone decreasingif the numerical solution remains monotone, TV = cst; otherwise, itdevelops new maxima and minima causing the TV to increaseif the approximation method is TVD, this cannot happen andtherefore the numerical solution remains monotone

TVD can be a stronger nonlinear stability condition than themonotonicity preserving condition





Drawback: Clipping phenomenon (illustrated with the linearadvection of a triangle-shaped initial condition)

The TV should increase by x between time-steps but a TVDscheme will not allow this clipping error (here this error is O(x)because it happens at a nonsmooth maximum, but for most smoothextrema it is O(x2)

)AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS




Summary of what should be known about TVD

in practice, most attempts at constructing a TVD scheme end upenforcing stronger nonlinearity stability conditions such as thepositivity condition discussed nextTVD implies monotonicity preservation: This is desirable whenmonotonicity preservation is too weak but less desirable whenmonotonicity preservation is too strong given that TVD can bestrongerTVD tends to cause clipping errors at extrema: In theory, clippingdoes not need to occur at every extrema since, for example, thelocal maximum could increase provided that a local maximumdecreased or a local minimum increased or a localmaximum-minimum pair disappeared somewhere else and may beonly moderate when it occurs: However, in practice, most TVDschemes clip all extrema to between first- and second-order accuracyin theory, TVD may allow large spurious oscillations but in practice itrarely does in any case, it does not allow the unbounded growthtype of instability




Positivity

Recall that the wave speed split form of a FT scheme is given by

un+1i = uni + C

+i+1/2(u

ni+1 uni ) Ci1/2(uni uni1)

C+i+1/2 0 and Ci+1/2 0

Suppose that a given FT numerical scheme can be written in wavespeed split form with

C+i+1/2 0, Ci+1/2 0 and C+i+1/2 + Ci+1/2 1 i (14)

Condition (14) above is called the positivity condition (also proposedfirst by Harten in 1983)

What is the connection between the positivity condition and thenonlinear stability of a scheme?




Positivity

The answer is: The positivity condition implies TVD

Example: FTFS applied to the nonlinear advection equationu

t+ a(u)

u

x= 0 is positive if 1 ani+1/2 0

Proof:

FTFS can be written in wave speed split form for the purpose ofnonlinear stability analysis as follows

un+1i = uni + C

+i+1/2(u

ni+1 uni ) Ci1/2(uni uni1)

where C+i+1/2 = ani+1/2 and Ci1/2 = 0hence C+i+1/2 + C

i+1/2 = ani+1/2 and therefore the condition (14)

becomes in this case 1 ani+1/2 0also, note that the positivity condition is in this case equivalent tothe CFL condition



Multidimensional Extensions

The extension to multiple dimensions of the computational part ofthe material covered in this chapter may be tedious in some casesbut is straightforward (except perhaps for the characteristic theory)

The expressions of the Euler equations in 2D and 3D can beobtained from Chapter 2 (as particular cases of the expression of theNavier-Stokes equations in 3D)

For simplicity, the focus is set here on the 2D Euler equations

W

t+Fxx

(W ) +Fyy

(W ) = 0




2D structured grid




W

t+Fxx

(W ) +Fyy

(W ) = 0

For the above 2D Euler equations, the equivalent of equation (2) ona 2D structured grid is

t

(W

t

)ni,j

= x(Fnxi+1/2,j Fnxi1/2,j ) y (Fnyi,j+1/2 Fnyi,j1/2 )

where

x =t

xi, y =

t

yjxi = xi+1/2,j xi1/2,j j , yj = yi,j+1/2 yi,j1/2 i

and Fxi+1/2,j and Fyi,j+1/2 are constructed exactly like fi+1/2 in 1D




For example, a 2D version of FTCS has the following conservativenumerical fluxes

Fnxi+1/2,j =1

2

(Fx(W ni+1,j) + Fx(W ni,j)

)

=1

2

(vx)i+1,j + (vx)i,j

(v2x )i+1,j + (v2x )i,j + pi+1,j + pi,j

(vxvy )i+1,j + (vxvy )i,j(Evx)i+1,j + (Evx)i,j + (pvx)i+1,j + (pvx)i,j

Fnyi,j+1/2 =

1

2

(Fy (W ni,j+1) + Fy (W ni,j)

)

=1

2

(vy )i,j+1 + (vy )i,j

(vxvy )i,j+1 + (vxvy )i,j(v2y )i,j+1 + (v

2y )i,j + pi,j+1 + pi,j

(Evy )i,j+1 + (Evy )i,j + (pvy )i,j+1 + (pvy )i,j


Conservative Finite Difference Methods in One DimensionForward, Backward, and Central Time MethodsDomain of Dependence and CFL ConditionLinear Stability AnalysisFormal, Global, and Local Order of AccuracyUpwind Schemes in One DimensionNonlinear Stability AnalysisMultidimensional Extensions

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