gas dynamics - uio

FYS-GEO 4500 Galen Gisler, Physics of Geological Processes, University of Oslo Autumn 2009

FYS-GEO 4500 12 Oct 2009

Before we start:Questions over the reading?

The problem set

Monday, 12 October 2009


Our schedule

date Topic Chapter in

LeVeque

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16

17 Aug 2009 Monday 13.15-15.00 introduction to conservation laws, Clawpack 1 & 2 & 5

24 Aug 2009 Monday 13.15-15.00 the Riemann problem, characteristics 3

28 Aug 2009 Friday 13.15-15.00 finite volume methods for linear systems 4

8 Sep 2009 Tuesday 13.15-15.00 high resolution methods 6

21 Sep 2009 Monday 13.15-15.00 boundary conditions, accuracy, variable coeff. 7,8, part of 9

29 Sep 2009 Tuesday 13.15-15.00 nonlinear conservation laws, finite volume methods 11 & 12

5 Oct 2009 Monday 13.15-15.00 nonlinear equations & systems 13, part of 14

12 Oct 2009 Monday 13.15-15.00 finite volume methods for nonlinear systems part of 14, 15

19 Oct 2009 Monday 13.15-15.00 varying flux functions, source terms part of 16, 17

26 Oct 2009 Monday 13.15-15.00 multidimensional hyperbolic problems & methods 18 & 19

2 Nov 2009 Monday 13.15-15.00 systems & applications; project planning 20 & 21

16 Nov 2009 Monday 13.15-15.00 applications: tsunamis, pockmarks, venting, impacts

23 Nov 2009 Monday 13.15-15.00 applications: volcanic jets, pyroclastic flows, lahars

30 Nov 2009 Monday 13.15-15.00 review; progress, problems & projects

7 Dec 2009 Monday 13.15-15.00 FINAL PROJECT REPORTS DUE



Gas Dynamics (Chapter 14 in Leveque)



!t+ (!u)

x= 0

!u( )t+ !u2 + p( )

x= 0

The barotropic equations and the shallow-water equations

The conservation laws for the barotropic system (i.e. with )

are exactly like the shallow water equations if we identify with h and use

the equation of state

Other barotropic forms include the isothermal equation of state

and the polytropic (or gamma-law) equation of state

But next we add the energy equation…

p = P(!)

!

p = P(!) =1

2g!2

p = P(!) = a2!,

p = P(!) = K!"



The Euler equations of gas dynamics

This is the full system of three conservation laws, for mass, momentum, and energy, for fully compressible gas dynamics:

where

The total energy is composed of internal energy plus kinetic energy,

and the system is completed by an equation of state .

The Jacobian has eigenvalues u–c, c, u+c where the speed of

sound is at constant entropy.

e = e(p,!)

!f (q)

c =dp

d!

q =

!

!u

E

"

#

$$$

%

&

''', f (q) =

!u

!u2+ p

u(E + p)

"

#

$$$

%

&

'''.

qt + f (q)x = 0,

E = !e+1

2!u2



The equation of state and associated relations for a polytropic gas

The ideal gas law:

The internal energy:

The enthalpy:

Relations between the specific heats:

number of molecules per unit mass Boltzmann’s constant

specific heat at constant volume number of degrees of freedom

specific heat at constant pressure adiabatic exponent

p

!= nkT

e = cvT =!

2nkT =

p

(" #1)$

h = e+p

!= cpT = 1+

"2

#$%

&'(nkT

cp ! cv = nk

" =cp

cv=

# + 2

#$%&

'()

*

+,

-,

k :n :

cp:

cv:

! :

! :



Entropy

In the system of Euler equations for gas dynamics, we have the advantage of having an explicit formula for entropy that we can use as an entropy condition.

The specific entropy s (i.e. entropy per unit mass) is given by the formula:

The additive constant is unimportant and may be omitted, since the important thing to keep track of is changes in entropy. In smooth flow, entropy is constant; at shocks it jumps to a higher value.

s = cvlog

p

!"

#$%

&'(+ constant



Primitive variables

It is often useful to examine the equivalent equations in directly observable “primitive” variables, rather than the conserved variables.

The Euler equations in primitive form for a polytropic gas:

in matrix notation:

!t+ u!

x+ !u

x= 0

ut+ uu

x+1

!px= 0

pt+ " pu

x+ up

x= 0

!

u

p

"

#

$$$

%

&

'''x

+

u ! 0

0 u1

!

0 ( p u

"

#

$$$$$

%

&

'''''

!

u

p

"

#

$$$

%

&

'''t

= 0

Then the eigenvalues and eigenvectors are:

where

is the speed of sound in the polytropic gas.

!1 = u " c !2 = u ! 3 = u + c

r1=

"# / c

1

"#c

$

%

&&&

'

(

)))

r2=

1

0

0

$

%

&&&

'

(

)))

r3=

# / c

1

#c

$

%

&&&

'

(

)))

c =! "

p



The Jacobian for the conservation laws

This (for the polytropic gas) is slightly more complex, though equivalent:

where is the total specific enthalpy.

And the eigenvalues and eigenvectors are:

!f (q) =

0 1 0

12 (" # 3)u2 (3# " )u " #1

12 (" #1)u3 # uH H # (" #1)u2 " u

$

%

&&&

'

(

)))

H =E + p

!= h +

1

2u2

!1 = u " c !2 = u ! 3 = u + c

r1=

1

u " c

H " uc

#

$

%%%

&

'

(((

r2=

1

u

1

2u2

#

$

%%%

&

'

(((

r3=

1

u + c

H + uc

#

$

%%%

&

'

(((



Genuine nonlinearity, linear degeneracy

Nonlinear systems of hyperbolic equations produce shocks and rarefaction waves; linear systems do not.

Nonlinear systems have integral curves and Hugoniot loci that diverge from one another; in linear systems these curves are identical.

But even in nonlinear systems, some waves can in fact act like linear waves in this sense.

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

2.5

-0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Shallow water equationsh

hu



Genuine nonlinearity, linear degeneracy

A wave (or field, we may say, referring to the collection of waves of the same family in all accessible space) is genuinely nonlinear if

Physically this means that the characteristics are either compressing or expanding.

The opposite case is linear degeneracy,

in this case the characteristics are parallel to one another.

!"pir

p(q) # 0!for all q

!"pir

p(q) = 0!for all q



The Euler equations have one linearly degenerate field

This is easiest to see in the eigensystem for the primitive equations. In this case the gradient operator is defined by

, so we find

And therefore

!1 = u " c !2 = u ! 3 = u + c

r1=

"# / c

1

"#c

$

%

&&&

'

(

)))

r2=

1

0

0

$

%

&&&

'

(

)))

r3=

# / c

1

#c

$

%

&&&

'

(

)))

! "

##$

##u##p

%

&

'''''''

(

)

*******

!"1 =

+c

2#

1

$c

2#

%

&

''''''

(

)

******

;!"2 =0

1

0

%

&

'''

(

)

***

;!"1 =

$c

2#

1

+c

2#

%

&

''''''

(

)

******

!"1ir1 = 1

2# +1( )

!"2 ir2 = 0

!" 3ir3 = 1

2# +1( )



The Riemann invariants for the polytropic gas

Thus, of the three eigenvectors, 1 and 3 represent waves that can become either rarefactions or shocks, while 2 is linearly degenerate and can only be a contact discontinuity.

For any simple wave (not a rarefaction or a shock), the Riemann invariants are constant along particle paths through the wave. These are, for the 3 waves:

1-wave: s, u +2c

! "1

2-wave: u, p

3-wave: s, u "2c

! "1



Now we can make more sense of the shock tube!

A closed tube filled with gas, separated by a membrane into sections with different densities.

The membrane is suddenly removed, and the gas starts moving from the high-density region into the lower density region.

Three waves develop: a shock wave, a contact discontinuity, and a rarefaction wave (or fan). The first two travel to the right, the third to the left.

At the shock, velocity, pressure and density are all discontinuous. At the contact, only density is discontinuous. In the rarefaction fan, all variables are continuous, but their derivatives are not.

membraneρh ρl

density

ρp

pressure

x

u

velocity waves

t

x

shock

contact

discontinuity

rarefaction

fan

ρh

ρl



The Riemann problem for 3 waves:

Now we have to solve for two intermediate states, not just one as before. And we have the rarefaction fan to deal with.

Note that only density changes across the contact discontinuity. This helps matters, in that we can use almost the same procedure as for the 2-equation shallow water set.

First we obtain and then we use separate conditions to determine and .

Qi

n

xi!1/2

tn+1

Qi!1

n

tn

!lul

pl

"

#

$$$

%

&

'''

!r

ur

pr

"

#

$$$

%

&

'''

!r

*

u*

p*

"

#

$$$$

%

&

''''

!l*

u*

p*

"

#

$$$$

%

&

''''

rarefaction contact shock

(p*,u

*)

!l

*!!!!!!!!!

r

*



The Riemann problem for 3 waves:

The 2-field is linearly degenerate. Across the contact u and p will be

constant and only ! will jump.

The strategy for solving the problem is to use the Hugoniot loci and integral

curves for the 1-field and 3-field, in the phase plane of u and p, in the same

way as for the shallow-water equations to obtain . Then we calculate the densities on either side of the contact. Finally we solve for the solution within the rarefaction fan.

Qi

n

xi!1/2

tn+1

Qi!1

n

tn

!lul

pl

"

#

$$$

%

&

'''

!r

ur

pr

"

#

$$$

%

&

'''

!r

*

u*

p*

"

#

$$$$

%

&

''''

!l*

u*

p*

"

#

$$$$

%

&

''''


(p*,u

*)



The general (exact) Riemann solver for the Euler equations for a polytropic gasAs before, we define functions

where .

We then require that , using an iterative procedure to find the intersection of the curves. The densities on either side of the contact will then be given by

u =! l p( ) =

ul +2cl

" #11# p / pl( )

" #1

2"$

%&'

() if p * pl

ul +2cl

2" " #1( )

1# p / pl

1+ + p / pl

$

%&&

'

())

if p , pl

-

.

//

0

//

u =!r p( ) =

ur #2cr

" #11# p / pr( )

" #1

2"$

%&'

() if p * pr

ur #2cr

2" " #1( )

1# p / pr

1+ + p / pr

$

%&&

'

())

if p , pr

-

.

//

0

//

! l pm( ) =!r pm( )

! =" +1

" #1

(p*,u

*)

!l*=1+ " p* / plp*/ pl + "

#$%

&'(!l ;!!!r

*=1+ " p* / prp*/ pr + "

#$%

&'(!r



Hugoniot loci and integral curves for the Euler Equations (polytropic gas)

-1.00

-0.75

-0.50

-0.25

0

0.25

0.50

0.75

1.00

0 0.5 1.0 1.5 2.0

Some integral curves (solid) and Hugoniot loci (dotted) for the Euler equations. Just as in the shallow water equations, these curves are close together in many places. An iterative solver can start from the intersection of the integral curves, which can be obtained explicitly.

These curves are

computed for "=1.4 and

densities !l= 3 and

!r=1. For lower " and

higher density contrasts, the curves spread further apart.

p

u



The 1-rarefaction to connect the

states ql and ql*. Again we take

advantage of the fact that the solution is a similarity solution, constant along the rays . And we also know that the 1-Riemann invariant is constant through a rarefaction wave:

We now have everything except the rarefaction:

Then we have that within the rarefaction wave, so we can rewrite the Riemann invariant as

and solve for u as a function of #.

Qi

n

xi!1/2

tn+1

Qi!1

n

tn


ql

ql*

qr

*

qr

! = x / t

u +2c

! "1= u

l+2c

l

! "1

! = "1 = u # c

u +2 u ! "( )# !1

= ul+2c

l

# !1,



The 1-rarefaction to connect the

states ql and ql*. Again we take

advantage of the fact that the solution is a similarity solution, constant along the rays . And we also know that the 1-Riemann invariant is constant through a rarefaction wave:

We now have everything except the rarefaction:

Then we have that within the rarefaction wave, so we can rewrite the Riemann invariant as

and solve for u as a function of #.

Qi

n

xi!1/2

tn+1

Qi!1

n

tn


ql

ql*

qr

*

qr

! = x / t

u +2c

! "1= u

l+2c

l

! "1

! = "1 = u # c

u +2 u ! "( )# !1

= ul+2c

l

# !1,

! =x

t



Continuing the solution within the rarefaction

Then since is constant, Qi

n

xi!1/2

tn+1

Qi!1

n

tn


ql

ql*

qr

*

qr

! =x

t

p / !"

c2= !

pl

"l!

#$%

&'("! )1

= u *( ) ) *( )2

and

! "( ) =!l

#

# plu(")$ "( )

2%&'

()*

1/(# $1)

p "( ) =pl

!l#

%&'

()*! "( )

#

u !( ) =" #1( )ul + 2 c

l+ !( )

" +1.

u +2 u ! "( )# !1

= ul+2c

l

# !1,

we solve for u as a function of #:

With the Riemann invariant




The exact solution is a perfect and complete structure and can be solved using iterative methods.


n

xi!1/2

tn+1

Qi!1

n

tn


ql

ql*

qr

*

qr

! =x

t

p / !"

c2= !

pl

"l!

#$%

&'("! )1

= u *( ) ) *( )2

and

! "( ) =!l

#

# plu(")$ "( )

2%&'

()*

1/(# $1)

p "( ) =pl

!l#

%&'

()*! "( )

#

u !( ) =" #1( )ul + 2 c

l+ !( )

" +1.

u +2 u ! "( )# !1

= ul+2c

l

# !1,







However, in practical computation all of this is not used; instead we will use approximate Riemann solvers.


n

xi!1/2

tn+1

Qi!1

n

tn


ql

ql*

qr

*

qr

! =x

t

p / !"

c2= !

pl

"l!

#$%

&'("! )1

= u *( ) ) *( )2

and

! "( ) =!l

#

# plu(")$ "( )

2%&'

()*

1/(# $1)

p "( ) =pl

!l#

%&'

()*! "( )

#

u !( ) =" #1( )ul + 2 c

l+ !( )

" +1.

u +2 u ! "( )# !1

= ul+2c

l

# !1,







However, in practical computation all of this is not used; instead we will use approximate Riemann solvers.

The exact solution is useful for verification, however.


n

xi!1/2

tn+1

Qi!1

n

tn


ql

ql*

qr

*

qr

! =x

t

p / !"

c2= !

pl

"l!

#$%

&'("! )1

= u *( ) ) *( )2

and

! "( ) =!l

#

# plu(")$ "( )

2%&'

()*

1/(# $1)

p "( ) =pl

!l#

%&'

()*! "( )

#

u !( ) =" #1( )ul + 2 c

l+ !( )

" +1.

u +2 u ! "( )# !1

= ul+2c

l

# !1,





Multifluid problems and other equations of state

The easiest multi-fluid case is when you have two ideal gases with different

values of ". Then you set up the Riemann problem at the interface between

the two fluids with right and left values for ". Things get complicated when

mixing occurs.

Other analytical or tabular equations of state can also be incorporated into a finite volume conservative scheme. Sage, for example, uses the Sesame library of tabular equations of state for lots of materials. The Sesame library, developed at Los Alamos National Laboratory, contains mostly industrial materials, with a few materials of geological interest.

Leveque gives lots of references to papers in which Riemann solvers are developed for other equations of state in his section 14.15. Some of these will be worth looking into for geological applications.



Dusty Gases, Anyone?

Marica Pelanti, a student of Randy Leveque, developed a code based on Clawpack for volcanic jets using a dusty gas model.

See*: Pelanti & Leveque, "High-Resolution Finite Volume Methods for Dusty Gas Jets and Plumes", SIAM J. Sci. Comput. 28 (2006) 1335-1360.

Their model was multifluid: Euler equations for the gas, and a pressure-less fluid for the dust, coupled together by drag and heat transfer.

Or… we could try a single-fluid dusty gas equation of state.

* http://www-roc.inria.fr/bang/Pelanti/



What happens when we add dust to gas?

The speed of acoustic waves in a general medium is,

which for an ideal gas is

Assuming the dust remains coupled to the gas (via Stokes drag), for a little loading the density increases without the pressure changing much.

Thus the speed of sound decreases. In fact, it does so dramatically.

cs=

!p!"

#$%

&'(isentropic

cs=

! p

"



An equation of state for a dusty gas*

One version of a dusty-gas equation of state is:

where K is the mass concentration and Z is the volume fraction of solid particles.

These are related through where is the particle solid density.

The speed of sound is then

where the ratio of specific heats for the mixture is

The specific heat of the dust particles is Csp and Cv is the specific heat at

constant volume of the gas.

p =(1! K )

(1! Z )"RT ,

cds =!p

"(1# Z )

! ="Cv (1# K )+CspK

Cv (1# K )+CspK.

K =Z!

s

!,

*from Vishwakarma, Nath, & Singh (2008), Physica Scripta 78 035402.http://stacks.iop.org/PhysScr/78/035402

!s



Density increases more rapidly than pressure as the dust content increases

ρ

Z (volume fraction of dust)

pressure right scaledensity left scale

p

2.0

10.0!

1.0

3.0

4.0

5.0

6.0

7.0

(g/cc)

0.00

0.01

0.10

1.00

1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+001E+00

(bar)

Assumptions:

spherical dust particles

150 ! radius, 2.5 g/cc density, specific heat 0.92 J/g K

air at density 1.204e-3 g/cc, temperature 293 K



Adding 2% dust (by volume) to air reduces the speed of sound to

50 m/s, and the mixture becomes nearly isothermal

0.6

0.8

1.0

1.2

1.4

1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+001E+00

0

100

200

300

400

1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+001E+00

speed of sound in the

dusty gas (m/s)

ratio of specific heats of the mixture

Γ

volume fraction occupied by dust particles

Assumptions:

spherical dust particles

150 ! in radius, 2.5 g/cc density, specific heat 0.92 J/g K

air at density 1.204e-3 g/cc, temperature 293 K


De Laval Nozzle

Invented by the Swedish inventor

Gustaf de Laval in 1897.

This nozzle is the basis of how jet

engines and rocket engines work.

The converging-diverging profile,

with a sufficient difference in

pressure between the reservoir and

the exhaust, results in a smooth

transition from subsonic to

supersonic flow.



How does a deLaval nozzle work?

v2

2+

!! "1

#$%

&'(p

)= constant along a streamline,

Bernoulli’s equation says that, for steady flow of a gas (ignoring gravity),

Suppose we have a reservoir at high pressure connected via a pipe to a medium at much lower pressure.

Then there is a maximum velocity at steady flow given by:

vmax

= c0

2

!0"1

where c0 and "0 refer to the thermodynamic conditions

in the reservoir (where v=0), and this maximum value is

obtained when the gas flows out into a vacuum (p=0).

=v

2

2+

cs

2

! "1 for an ideal gas.




Using where c is the local sound speed

throughout the system, we get

Euler’s equation gives us the relation between v and ! along a streamline:

vdv =dp

!.

dp = c2!,

d !v( )dv

= ! 1"v2

c2

#$%

&'(,

indicating that the mass flux !v has a maximum where v is

equal to the local sound speed. By continuity, this maximum

holds at the narrowest point of the pipe, where v itself is

equal to its maximum.

For the isentropic case, the flux is

!v =p

p0

"#$

%&'

1/(2(( )1

p0!01)

p

p0

"#$

%&'

( )1( )/(*

+,,

-

.//




The maximum discharge rate is

reached, if ever, at the narrowest

point of a nozzle.

If the pressure at this point is less

than 0.53 p0 (in air) or less than 0.6

p0 (in a 2% dusty gas), the flow

speed equals the sound speed at

that point. In the diverging portion,

it becomes supersonic without

passing through a shock.

In a dusty gas it is easier to get to

supersonic flow both because the

local sound speed is lower and

because the maximum discharge is

reached with a lower pressure

drop.

!v =p

p0

"#$

%&'

1/(2(( )1

p0!01)

p

p0

"#$

%&'

( )1( )/(*

+,,

-

.//,

0

0.175

0.350

0.525

0.700

0 0.25 0.50 0.75 1.00

! = 1.4

! = 1.01

p / p0

!v

!0c0


Vents, kimberlite pipes, and geysers may be natural deLaval nozzles for a dusty gas

1 cm

Hyd

roth

erm

al a

lte

ratio

na

nd

fra

ctu

rin

g

Bore

hole

Sedimentpipes

Slumping

Zeolite

Vent sandstone(Facies unit I)

Sediment breccia(Facies unit III)

Zeolite-cementedsandstone (Faciesunit II)

100 m

100 m

200 m

300 m

0 m

Facies unit II

Erosion profile



Shocks can be stronger in dusty gases

The maximum ratio of upstream to downstream densities across a shock is (see

Landau & Lifshitz, Fluid Dynamics)

For a diatomic gas (like air),

For a dusty gas, can be arbitrarily large. At

!u

!d

=" +1

" #1.

! = 1.4, so "u

"d

= 6.

! "1, so #u

#d

! = 1.01, "u

"d

= 201.



The Mie-Grüneisen-Lemons Multiphase EOS

This is an equation of state proposed by Don Lemons based on Mie-Grüneisen theory for substances that undergo relatively simple phase transitions from solid to liquid to gas.

In common with the van der Waals EOS (see Leveque, ch 14.15 and 16.3.2), this EOS results in loss of hyperbolicity unless Maxwell constructions are used.

This EOS has substance-specific constants

p = !cv"T +

3B

n # m( )

""0

$%&

'()

n+3

3

#""0

$%&

'()

m+3

3*

+

,,,

-

.

///

e = cvT +

9B

"0n # m( )

1

n

""0

$%&

'()

n

3

#1

m

""0

$%&

'()

m

3

#1

n+1

m

*

+

,,,

-

.

///

n,m,!0,B,"



Finite Volume Methods for Nonlinear Systems

(Chapter 15 in Leveque)



Once again, we extend from what we’ve learned for linear systems of equations

We intend to solve the nonlinear conservation law

using a method that is in conservative form:

and yielding a weak solution to this conservation law. To get the correct weak solution we must use an appropriate entropy condition.

qt + f (q)x = 0

Qi

n+1=Q

i

n!"t

"xFi+1/2

n! F

i!1/2

n( )



Recall Godunov’s method:

Given a set of cell quantities at time n:

1. Solve the Riemann problem at to obtain

2. Define the flux:

3. Apply the flux differencing formula:

This will work for any general system of conservation laws. Only the formulation of the Riemann problem itself changes with the system.

Fi!1/2n

= f Qi!1/2

"( )

Qi

n

Qi!1/2

"= q

"(Qi!1

n,Qi

n)x

i!1/2

Qi

n+1=Q

i

n!"t

"xFi+1/2

n! F

i!1/2

n( )



In terms of the REA scheme we have discussed

1. Reconstruct a piece-wise linear function from the cell averages.

with the property that TV(q) " TV(Q)

2. Evolve the hyperbolic equation (approximately) with this function to obtain a later-time function, by solving Riemann problems at the interfaces.

3. Average this function over each grid cell to obtain new cell averages.

!qn(x,t

n+1)

The reconstruction step depends on the slope limiter that is chosen, and should be subject to TVD constraints. The other two steps do not affect TVD.

qn(x,tn ) =Qi

n+! i

n(x " xi ) for x in cell i

Qi

n+1=1

!x!qn(x,tn+1)

xi"1/2

xi+1/2

# dx

evolve

reconstruct

average



Recall what TVD means:

Monotonicity preserving methods:

If a grid function that is initially monotone, i.e.

remains monotone at the next time:

then the method is monotonicity preserving.

Total Variation Diminishing (TVD) methods:

Define the total variation of a grid function Q as:

A method is Total Variation Diminishing if

If Qn is monotone, then so is Qn+1, and no spurious oscillations are

generated.

This gives a form of stability necessary for proving convergence, also for nonlinear conservation laws.

TV(Q) = Qi!Q

i!1

grid

"

TV(Qn+1) ! TV(Q

n)

Qi

n!Q

i"1

n for all i at step n

Qi

n+1!Q

i"1

n+1 for all i at step n +1



For advection, the REA algorithm gives us:

We update the advection equation by

where the slope is given by

where is the flux limiter function.

Choices for the flux limiter are:

Qi

n+1=Q

i

n!u"t

"xQi

n!Q

i!1

n( )!1

2

u"t

"x"x ! u"t( ) # i

n!#

i!1

n( )

!i

n=

Qi+1

n "Qi

n

#x$%&

'()*i

n

!

!(") = 0

!(") = 1

!(") = "

!(") =1

2(1+")

!(") = minmod(1,")

!(") = max(0,min(1,2"),min(2,"))

!(") = max(0,min((1+") / 2,2,2"))

!(") =(" + " )

(1+ " )

upwind:

Lax-Wendroff:

Beam-Warming:

Fromm:

minmod:

superbee:

MC:

van Leer:

!i

n=Qi

n"Q

i"1

n

Qi+1

n"Q

i

n



Slope limiters and flux limiters

The slope limiter formula for advection is:

The flux limiter formulation for advection is:

with the flux:

Qi

n+1=Q

i

n!u"t

"xQi

n!Q

i!1

n( )!1

2

u"t

"x"x ! u"t( ) # i

n!#

i!1

n( )

Qi

n+1=Q

i

n!"t

"xFi+1/2

n! F

i!1/2

n( )

Fi!1/2

n= uQ

i!1

n+1

2u("x ! u"t)#

i!1

n



Wave limiters:

Let:

Upwind formula:

Lax-Wendroff formula:

High-resolution method:

where

xi!1/2

Qi

n

Wi!1/2

Qi!1

n

Wi!1/2 =Qi

n!Q

i!1

n

Qi

n+1=Q

i

n!u"t

"xW

i!1/2

Qi

n+1=Q

i

n!u"t

"xW

i!1/2!"t

"xFi+1/2 ! Fi!1/2( )

Fi!1/2 =

1

21!

u"t"x

#$%

&'(uW

i!1/2

!Fi!1/2 =

1

21! u

"t"x

#$%

&'(uW" i!1/2 ,

W!

i!1/2 = "i!1/2Wi!1/2 .



Extension to linear systems

Approach 1:

Diagonalise the system to

Apply the scalar algorithm to each component separately.

Approach 2:

Solve the linear Riemann problem to decompose into a number of waves.

Apply a wave limiter to each wave.

These approaches are equivalent.

But note that it is important to apply the limiters to the waves rather than to the original variables.

qt+ !q

x= 0

Qi

n!Q

i!1

n

xi!1/2

xi+1/2

Qi

nQi+1

n

Wi!1/22

Wi!1/23

Wi+1/21

Qi!1

n

Wi!1/21



Wave propagation methods

Solving the Riemann problem between cells i and i+1 gives the waves

and speeds . In the nonlinear case, an approximate solution is used.

These waves update the neighbouring cell averages via fluctuations, depending on sign of .

The waves also give the decomposition of the slopes

Apply the limiter to each wave to obtain

Use the limited waves in the second-order correction terms.

xi!1/2

xi+1/2

Qi

nQi+1

n

Wi!1/22

Wi!1/23

Wi+1/21

Qi!1

n

Wi!1/21

Qi !Qi!1= Wi!1/2

p

p=1

m

" ,

si!1/2

p

si!1/2

p

Qi !Qi!1

"x=

1

"xWi!1/2

p

p=1

m

# ,

W!

i!1/2 = "i!1/2Wi!1/2 .



High-resolution wave-propagation scheme

The fluctuation notation is more useful in the nonlinear case:

where

represents the limited version of .

This is obtained by comparing with where

Fi!1/2n

=1

2si!1/2p

1!"t"x

si!1/2p#

$%&'(W! i!1/2

p

p=1

m

) Qi

n+1=Q

i

n!"t

"xA

!"Q

i+1/2 +A+"Q

i!1/2( )!"t

"xFi+1/2 ! Fi!1/2( )

W!

i!1/2

p

Wi!1/2p

Wi!1/2p

Wl!1/2p

l =i !1 if si!1/2

p> 0

i +1 if si!1/2

p< 0

"#$%



Wave limiters for a system

is split into waves

We replace these waves with the limited versions

where , and .

Note that if then .

In the scalar case this reduces to

xi!1/2

xi+1/2

Qi

nQi+1

n

Wi!1/22

Wi!1/23

Wi+1/21

Qi!1

n

Wi!1/21

Qi !Qi!1 Wi!1/2

p= " i!1/2

pri!1/2

p,

!Wi!1/2

p= "(#

i!1/2

p)W

i!1/2

p.

!i"1/2p

=Wi"1/2

p#Wl"1/2

p

Wi"1/2p

#Wi"1/2p

l =i !1 if si!1/2

p> 0

i +1 if si!1/2

p< 0

"#$%

ri!1/2p

= rl!1/2p

!i"1/2p

=# l"1/2

p

# i"1/2

p

!i"1/2p

=Wl"1/2

p

Wi"1/2p

=Ql "Ql"1

Qi "Qi"1



The exact Riemann solver for the nonlinear problem is expensive, and most of it is not necessary!

These are useful for exact solutions of certain problems.

But in simulations we only need the solution at the cell interface!

Qi

n

xi!1/2

tn+1

Qi!1

n

tn

ql

ql*

qr

*

qr

u =! l p( ) =

ul +2cl

" #11# p / pl( )

" #1

2"$

%&'

() if p * pl

ul +2cl

2" " #1( )

1# p / pl

1+ + p / pl

$

%&&

'

())

if p , pl

-

.

//

0

//

u =!r p( ) =

ur #2cr

" #11# p / pr( )

" #1

2"$

%&'

() if p * pr

ur #2cr

2" " #1( )

1# p / pr

1+ + p / pr

$

%&&

'

())

if p , pr

-

.

//

0

//

! l pm( ) =!r pm( ) (p*,u

*) !l

*=1+ " p* / plp*/ pl + "

#$%

&'(!l ;!!!r

*=1+ " p* / prp*/ pr + "

#$%

&'(!rSolving yields , then

In the rarefaction fan,

! "( ) =!l

#

# plu(")$ "( )

2%&'

()*

1/(# $1)

p "( ) =pl

!l#

%&'

()*! "( )

#

u !( ) =" #1( )ul + 2 c

l+ !( )

" +1.



The exact Riemann solver for the nonlinear problem is expensive, and most of it is not necessary!

These are useful for exact solutions of certain problems.

But in simulations we only need the solution at the cell interface!

Qi

n

xi!1/2

tn+1

Qi!1

n

tn

ql

ql*

qr

*

qr

u =! l p( ) =

ul +2cl

" #11# p / pl( )

" #1

2"$

%&'

() if p * pl

ul +2cl

2" " #1( )

1# p / pl

1+ + p / pl

$

%&&

'

())

if p , pl

-

.

//

0

//

u =!r p( ) =

ur #2cr

" #11# p / pr( )

" #1

2"$

%&'

() if p * pr

ur #2cr

2" " #1( )

1# p / pr

1+ + p / pr

$

%&&

'

())

if p , pr

-

.

//

0

//

! l pm( ) =!r pm( ) (p*,u

*) !l

*=1+ " p* / plp*/ pl + "

#$%

&'(!l ;!!!r

*=1+ " p* / prp*/ pr + "

#$%

&'(!rSolving yields , then

In the rarefaction fan,

! "( ) =!l

#

# plu(")$ "( )

2%&'

()*

1/(# $1)

p "( ) =pl

!l#

%&'

()*! "( )

#

u !( ) =" #1( )ul + 2 c

l+ !( )

" +1.



But we do need to know where the cell interface lies with respect to the waves, and compute accordingly

Qi

n

xi!1/2

Qi!1

n

ql

ql*

qr

*

qr

Qi

n

xi!1/2

Qi!1

n

ql

ql*

qr

*

qr

Qi

n

xi!1/2

Qi!1

n

ql

ql* q

r

*

qr

Qi

n

xi!1/2

Qi!1

n

ql

ql* q

r

*

qr

Qi

n

xi!1/2

Qi!1

n

ql

ql*

qr

*

qr



Wave interactions

Waves that interact with each other within a cell will not change the cell-interface values on the next time step provided the Courant number is less than 1.

Qi

n

xi!1/2

Qi!1

nQi+1

n

xi+1/2



Riemann solvers in CLAWPACK

In CLAWPACK, the hyperbolic problem is specified by providing a Riemann

solver with

Input: the value of q in each grid cell

Output: the solution to the Riemann problem at each cell interface:

The Waves for a system of m equations

The Speeds

The Fluctuations , for high-resolution corrections

Because the problem is solve entirely using Riemann solvers, you won’t see anything in the code that resembles the original system of partial differential equations.

Wp, p = 1,2,…,m

sp, p = 1,2,…,m

A±!Q



Wave propagation for nonlinear systems

An approximate Riemann solver is typically used to get the wave decomposition

where the wave propagates at a speed .

If we define as a linearised approximation to

valid in the neighbourhood of (Qi, Qi-1 ),

then we can solve the simpler linear Riemann problem at that cell interface for the linearised equation:

to obtain

Qi !Qi!1= Wi!1/2

p

p=1

m

" ,

Wi!1/2p s

i!1/2

p

qt + Ai!1/2qx = 0,

Ai!1/2

= A(Qi

n,Q

i!1

n) !f (q)

Wi!1/2

p= "

i!1/2

pri!1/2

p, s

i!1/2

p= #

i!1/2

p.



Approximate Riemann Solvers

Approximate the true Riemann solution by a set of waves consisting of finite jumps propagating at constant speeds (as in the linear case).

Use a local linearisation: replace by

where

Then decompose

to obtain the waves with speeds

But how do we chose

qt + f (q)x = 0 qt + Ai!1/2qx = 0,

A = A(ql ,qr ) ! "f (qave ).

ql ! qr = "prp

p=1

m

#

Wp= !

prp s

p=

ˆ"p.

A?



Approximate Riemann Solvers

Properties desired for

must be diagonalisable with real eigenvalues

With these properties, the method will be conservative, it will give the right answer across shocks (why?), and it is a good approximation for smooth flow.

We could take where is a suitable average

of . We’ll use basically this approach.

Or we could take

A :

Ai!1/2" #f (q)!!!as Qi!1

,Qi " q

Ai!1/2 Qi !Qi!1( ) = s Qi !Qi!1( ) = f Qi( )! f Qi!1( )

A

Ai!1/2 = "f Qi!1/2( ) Qi!1/2

Qi!1,Q

i( )

Ai!1/2 =1

2"f Qi!1( ) + "f Qi( )#$ %&.



Example: Roe solver for shallow-water equations

= depth

= bulk speed, varies only with x

Conservation of mass and momentum gives the system:

which has the Jacobian matrix:

qt + f (q)x =h

hu

!

"#

$

%&t

+

hu

hu2+1

2gh

2

!

"

###

$

%

&&&x

= 0.

!f (q) =0 1

"u2+ gh 2u

#

$%%

&

'((, )=u ± gh.

h = height of wave above bottom

u = speed of bulk water motion! = water density

h(x,t)

u(x,t)



Roe solver for Shallow Water

Given define .

Then if is defined as the Jacobian matrix evaluated at the special state

we find that:

the Roe conditions are satisfied,

an isolated shock is modelled well,

and the wave propagation algorithm is conservative.

If we use limited waves, we obtain high-resolution methods as before.

hl,u

l,h

r,u

r, h =

hl+ h

r

2, u =

hlul+ h

rur

hl+ h

r

A

q = (h ,hu),




Given define .


we find that:

the Roe conditions are satisfied,

an isolated shock is modelled well,

and the wave propagation algorithm is conservative.

If we use limited waves, we obtain high-resolution methods as before.

hl,u

l,h

r,u

r, h =

hl+ h

r

2, u =

hlul+ h

rur

hl+ h

r

A

q = (h ,hu),

“Roe average”




Given define .


the eigenvalues of are:

and the eigenvectors are:

hl,u

l,h

r,u

r, h =

hl+ h

r

2, u =

hlul+ h

rur

hl+ h

r

A = !f (q)

!1= u " c, !

2= u " c, c = gh ,

r1=

1

u ! c

"

#$

%

&', r

2=

1

u + c

"

#$

%

&',

A

q = (h ,hu),



The Shallow-water Riemann solver in Clawpack is a Roe solver

“Roe average”

from: $CLAW/book/chap13/swhump1/rp1sw.f

h =hl+ h

r

2, u =

hlul+ h

rur

hl+ h

r



The Harten-Lax-van Leer (HLL) Solver

This solver uses only 2 waves with

s1 = minimum characteristic speed

s2 = maximum characteristic speed

Write

where the middle state is uniquely determined by the conservation requirement:

Modifications of this include positivity constraints and the addition of a third wave.

W1=Q

*! ql , W

2= qr !Q

*

Q*

s1W

1+ s

2W

2= f (qr )! f (ql )

"Q*=f (qr )! f (ql )! s

2qr + s

1ql

s1! s

2



f-wave approximate Riemann solver

Instead of splitting Q into waves, we might consider splitting the flux f into

“waves” :

It turns out this is useful for spatially varying flux functions, i.e.

with applications, for example, in:

wave propagation in heterogeneous nonlinear media,

flow in heterogeneous porous media,

traffic flow with varying road conditions,

conservation laws on curved manifolds,

and certain kinds of source terms.

f (Qi )! f (Qi!1) = Zi!1/2p

p=1

Mw

"

(Mw! w)

Mw

qt + f (q, x)x = 0,



Flux-based wave decomposition (f-waves)

Choose wave forms rp (for example, eigenvectors of the Jacobian on each

side).

Then decompose the flux difference:

fr (qr )! fl (ql ) = " prp

p=1

m

# = Zp

p=1

m

#

qr

W2

W3

ql

W1

qr

*ql*

ql + fl (q)x = 0 qr + fr (q)x = 0



Wave propagation algorithm using waves

In the standard version:

Qi

n+1=Q

i

n!"t

"xA

!"Q

i+1/2 +A+"Q

i!1/2( )!"t

"xFi+1/2 ! Fi!1/2( )

Qi !Qi!1 = Wi!1/2p

p=1

m

"

A!#Qi+1/2 = si+1/2

p( )!Wi+1/2

p

p

"

A+#Qi!1/2 = si!1/2

p( )+

Wi!1/2p

p

"

!Fi!1/2 =1

2si!1/2p

1!#t#x

si!1/2p$

%&'()!Wi!1/2p

p=1

m

"

*

+

,,,,,

-

,,,,,



Wave propagation algorithm using f-waves

Using f-waves:

Qi

n+1=Q

i

n!"t

"xA

!"Q

i+1/2 +A+"Q

i!1/2( )!"t

"xFi+1/2 ! Fi!1/2( )

fi (Qi )! fi!1(Qi!1) = Zi!1/2p

p=1

m

"

A!#Qi+1/2 = Zi+1/2

p

p:si!1/2p

<0

"

A+#Qi!1/2 = Zi!1/2

p

p:si!1/2p

>0

"

!Fi!1/2 =1

2sgn si!1/2

p( ) 1!#t#x

si!1/2p$

%&'()!Zi!1/2p

p=1

m

"

*

+

,,,,,

-

,,,,,



f-wave approximate Riemann solver

Let be any averaged Jacobian matrix, for example:

Use eigenvectors of to do f-wave splitting.

Then , so the method is conservative.

If is the Roe average, then this is equivalent to the normal Roe Riemann solver, and

A

A = !fql + qr

2

"#$

%&'

A!"Qi+1/2 +A

+"Qi!1/2 = f (Qi )! f (Qi!1)

A

A

Zp= s

pW

p.



Assignment for next time

Read Chapter 14 and Chapter 15.

Write (in Fortran) an approximate Riemann solver for the Euler equations using the Roe average. Test it on the shock tube problem, or (optionally) on the Woodward-Collella blast-wave problem. Use the shallow-water Riemann solver as a guide.

Hand in your code and results to me by Monday 19 October.


gas dynamics - uio

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