ESAIM Proceedings Vol Septembre

Actes du eme Congres

dAnalyse Numerique CANum

URL httpwwwemathfrprocVol

Riemann Solvers for some Hyp erb olic Problems

with a Source Term

Alain Yves LE ROUX

Resume Dans de nombreux mo deles de mecanique des uides le terme source

corresp ond auneinteraction de lecoulementavec le milieu environnant la geometrie

ou certaines forces internes telles que la friction la gravite la force de Coriolis etc

Dans les problemes denvironnement cette interaction est souvent un terme dominant

qui determine les equilibres etats au rep os Lors des simulations numeriques il est

evidemmenttres imp ortant de repro duire parfaitementcesetats dequilibre et de les

mainteniral equilibre au cours du temps Il est surtout imp ortantdeviter que la p erte

de lequilibre au niveau numerique vienne destabiliser et mo dier de facon signicative

la simulation Dans ce papier cette question est analysee a partir dexemples concrets

et des solveurs appropries dits solveurs equilibres sont prop oses

Mots cles

Problemes denvironnement stabilitedeschemas numeriques p our les systemes hy

p erb oliques solveurs equilibres volumes nis Go dunov

Abstract In many problems of uid dynamics the source term corresp onds to the

interaction with the surrounding medium including the geometry and some internal

forces friction gravity Coriolis acceleration etc In environmental problems this

interaction is often a very dominanttermwhichcharacterizes the well balanced states

Inanumerical simulation it is obviously imp ortant to b e able to repro duce these

equilibria and maintain them for a large numb er of timesteps or at least to avoid a

stable equilibrium to b e disturb ed and destabilized by an unsuitable metho d This is

the matter discussed in the present pap er

Keywords Problems in environmental science stabilityofnumerical schemes for

hyp erb olic systems well balanced solvers nite volumes Go dunov

AMS Sub ject Classication M M

Some examples

We rst consider the following one dimension mo del

u uu a

t x

Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1999047

Solvers for some Hyp erb olic Problems with a Source Term

where a is a given function for example

a

ax

Kx

with given constants a OK We start with the following initial data for a given

small

for x

ux

maxa ax forx

The solution of this problem is obviously the stationary solution since the ux term uu

a is always zero either u or u a a constant

x

A usual waytosolvenumerically this problem is to use a splitting technique that

is to cut into two steps which will b e run consecutively at each timestep during the

computation These two steps are

u

S tep u

t x



S tep u uax

t

Let us consider a p oint x close to Wehave u andu and from

x

the solution u will increase and moveforwards This can create a value u on the





top of a at x which will not b e corrected by since a x on the top Wecan

exp ect that the solution will still move rightwards and b ecome a sho ckwave This sho ck

wavewillhave the following form

u a ax for xxt



ux t

for x xt

where x xt corresp onds to the sho ck path whose equation is given by the Rankine

Hugoniot relation



u a ax x t



a



starting from x For a large xwehave x t and a jump close to a which

means a large one This example shows how a tiny error at the bad place will generate

a large error that cannot b e controled later It is a very dicult task to prevent this

phenomenon by using a splitted numerical metho d corresp onding to a discretization of

A p ossible wayistouseavery small meshsize One of the aims of this pap er is

to prop ose other schemes whichmaintain the equilibria An example of numerical scheme

for this mo del will b e presented in Section It is not a splitted metho d

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

Some physical mo dels

Hydraulics

We can consider the ab ove function axgiven in as the prole of a dam with a water

reservoir on the x We consider the following socalled ShallowWater one dimension

mo del

h hu

t x

hu hu ghh a

t x x

where h is the heigth of water u the velo cityandg the gravity constant We

can notice that this mo del is relevanteven for deep er ows as long as the velo city stays

homogeneous on the thickness of the water layer otherwise this mo del corresp onds to a

damp ed mo del since the velo cityisaveraged which means reduced as seen easily byusing

the CauchySchwartz inequality

We start with the following initial data

for x

hx

maxZ ax for x

ux

where Z corresp onds to the water level For Z a we see clearly that the solution is

the stationary one since the water will not climbby itself up to the top of the dam

Now by using a splitting technique as the following one

h hu

t x

Step



h

hu hu g

x t

h

t

Step



hu ghax

t

We can notice that the rst step will move the water rightwards from the left hand side

of the dam If Z is close to a with Z a a p ositive heigth of water will app ear on

the top of the dam will provoke a ux forwards and a ow dropping down the hill This

juj

p

larger that one this is to b e ow will b e a torrential one with a Froude number

gh

compared to a sup ersonic owinhydro dynamics characterized by a thin layer of water

with a large velo city In such a situation a numerical scheme can pro duce a negative

value for h and download the computer Obviously this cannot b e prevented by the



second step since a x is zero on the top of the dam and negative rightwards which will

accelerate the ow

ESAIM Proc Vol Septembre

Riemann Solvers for some Hyp erb olic Problems with a Source Term

Atmosphere

We consider the Euler equations with a constant gravityforaonevertical dimension

The ground corresp onds to x The mo del reads

u

t x

u u p g

t x

e e pu g u

t x

where is the density u is the velo city g the gravity constant p the pressure and e the

total energy The two last parameters p and e are linked by the internal energy I through

the formulae

u

p I e I

where is the adiabatic constant of the air From and we can reduce

to the following equation on the pressure

p up pu

t x x

where no source term is present This shows that the only non homogeneous equation in

the system is

We consider now an equilibrium at rest which means that u and all the time

derivatives are zero The equations or are reduced to zero zero and

is reduced to the single equation

I g

x

whichinvolves two parameters We can construct an innity of solutions Among them

are these two particular ones for a given state I at the ground level

gx



I



I I e

and

gx

I I

The rst one corresp onds to a strong tornado since the larger part of the air will b e pulled

I

up The second one will pro duce a negativeinternal energy at the level x

g

which has no physical meaning As a matter of fact an admissible stationary solution

will corresp ond to exp erimental measurements and it is really dicult to preserve suchan

equilibrium numerically A splitting technique will simulate a catastrophic b ehaviour of

the atmosphere far stronger than any phenomenon we use to mo delize in environmental

science A suitable metho d is presented in the next section This metho d do es not

corresp ond to a new scheme but to a new formulation of the mo del whichbecomesa

homogeneous one

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

Awell balanced metho d for the Atmosphere mo del

We consider the mo del of atmosphere describ ed in the previous section that is

and or We recall that g is a constant The transp ort equation expresses

that the vector u isdivergence free in the x t set Hence there exists a p otential

denoted by x t such that

u

x t

which leads to the equation

u

t x

We set

q g

and by using we get that q satises a transp ort equation of velo city uThatway

wehave got a new formulation of the mo del

q q u

t x

u

t x

u u p q

t x

e q u pu

t x

which is a homogeneous system of conservation laws It is still an hyp erb olic problem

whose eigenvalues of the ux are

r

p

u c u double u c with c

Moreover the eigenspace asso ciated with the double eigenvalue u has two dimensions

so that the system has the same prop erties as a strictly hyp erb olic one We nd again

the same Riemann invariant as for the homogenous version of and and

the same sho ck conditions derived from the Rankine Hugoniot relations for the three

parameters u and pOnashockwavewe get that q is continuous no jump The only

dierence comes from the contact discontinuities where the continuous variables are now

the velo city u and the total pressure p q instead of u and p in the previous mo del This

allows to relax the pressure and help the preservation of the equilibria

Wecannow solve this system by using any classical scheme For exemple the

Go dunov metho d works very well and preserve the equilibria even when starting from

exp erimental measured data More the numerical tests using the Go dunovscheme show

that the stability of an initial equilibrium will b e recovered after a strong lo cal p erturbation

The only mo dications in the Go dunovscheme are the following

the is mo died by including the balance p q constant on the

contact discontinuities

the source term is taken in account together with the ux term since it b ecomes

a ux term itself in the new mo del

And we add this essential remark the CFL condition is not p erturb ed This is

another very imp ortantadvantage on the splitting technique

ESAIM Proc Vol Septembre

Riemann Solvers for some Hyp erb olic Problems with a Source Term

Awell balanced scheme for the scalar mo del

We try to adapt the previous idea for the scalar equation

u f u g uax

t x



We supp ose for convenience that f g C Rf andthata is a given function

A rst naive idea is to intro duce a function q suchthatq g uax in order

x

to write under the form

u f u q

t x

However we get no homogeneous equation for q in the general case and this idea is to b e

rejected

Another idea is to intro duce a primitive function Axof ax and give the

form



f u



u A x u g u

x t

g u

whichis

u g u w

t x

by writing successively



f u



and w u Ax u

g u

We get



f u



w u u g u w

t t x

g u

that is



w f u w

t x

Nowwehave a homogenous system made of and whichpresents however some

drawbacks it is not a conservative one and a new eigen value has b een created

On the other hand wegetfrom that w has the b ehaviour of a Riemann invariant



along the characteristics of velo city f u We shall use it to solve the

noticing that a wide class of function Axmay b e used including the class of constant

piecewise functions In the construction of the metho d we shall approach the function

Axby a constant piecewise function which is constantoneach cell and use to

build the Riemann solver

Let x b e the meshsize and t the timestep Wesetx ix for i Zand

i

M x x The approximate solution at time t nt on the cell M will b e denoted

i i i n i

n

u We build a constant piecewise function A dened by

x

i

Z

x

i

A x A a d for x M

x i i

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

and set for each cell M

i

n n

w u A

i

i i

By using one gets that w is constant across eachinterface x x and travelling

i

  

forwards along the characteristic x tf uwith f u The Riemann problem near

the interface x x is made of the equation and the starting data

i

n

w for x x

i

i

w x t

n

n

forxx w

i

i

 

We denote by w respu the trace of w respu along the right side of the interface

i i

 n

x x Wehave w w which leads to

i

i i

 n

u A A u

i i

i i

Since may b e discontinous at the ro ots of g and continous in the inside of the intervals

 n

limited by these ro ots wetake of course u in the interval where u is lying

i i

The Go dunovscheme which is here close to the decentered scheme reads

n

n  n

u u r f u f u

i i i

i

t

We get it immediately byintegrating where r

x



u f u g u A x

t x

x



on the set M t t and noticing that A is zero in the inside of the cell M

i n n i

x

The numerical metho d is made of and and works under the usual CFL

condition for the decentered scheme that is



r max f u



One can prove L and BV estimates and the convergence towards the entropy condition

see

This metho d works obviously in the case ax and a given g u and we notice

d x

that the constant a is approached by the Dirac comb E x where E states

dx x

for the integer part It works obviously in the case g u a axtoo

This metho d has b een tested on the example for a small and the

initial equilibrium was p erfectly resp ected indep endently on Itwas not the case for

the usual decentered scheme coupled with a splitting technique with the same meshsize

By using a very small meshsize connected with the size of we can however exp ect this

splitting metho d to work for there is a well known result of convergence

ESAIM Proc Vol Septembre

Riemann Solvers for some Hyp erb olic Problems with a Source Term

The ShallowWater mo del

This section is based on the results obtained in

We consider a water owover either a river b ed or the surrounding ground when

o o ding or the b ottom of the o cean for a coastal ow We rst consider a one dimension

mo del and denote by a ax the b ottom elevation by h the heigth of water h

corresp onds to a dry ground by g the gravity constant and by u the water velo city when

h The mo del is the following

h hu

t x

hu hu gh h a

t x x

whichisknown as the ShallowWater mo del thought it still works for deep water

to o in many cases This mo del can b e built from the uncompressible Euler equations by

integrating over the thickness of the water layer Wetake the average velo city

Z

ah

ux t ux z t dz

h

a

whereu is the velo city in the Euler mo del The equation is derived immediately from

the uncompressibilityhyp othesis in the Euler mo del and the equation is obtained

by using the hydrostatic pressure that is

p

g

z

and the averaged velo citygiven by instead ofu Note that this last approximation

can b e seen as the contribution of a friction phenomenon since from the CauchySchwartz

inequalitywehave

Z Z

ah ah

ux z t dz h jux z tj dz h u

a a

The equations corresp ond to a hyp erb olic system with the characteristics

p

juj

velo cities u c where c gh is called the wave celerity The rate is called the



c

Froude number For juj c we get the critical velo cityAFroude numb er greater than

corresp onds to a sup ercritic mo de also called torrential mo de and a Froude numb er less

than corresp onds to a sub critic mo de also called uvial mo de

As for the scalar case the b ottom prole ax will b e approached by a function a

x

constant on each cell We get that way a at b ottom in each cell that is a homogeneous

conservative form of in the inside of eachcell This will allow to use the

divergence free argument to p erform the pro jections and get easily the Go dunovscheme

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

with the usual CFL condition once the Riemann problem is solved on the two sides of the

cell

The Riemann problem is made of the equations and the initial data

h u for x

d d

hx ux

h u for x

g g

We consider the case of the following b ottom prole

a for x

g

x

a x a a a for x

g d g

a for x

d

for a given detinated to go to zero In and the data a h u and

g g g

a h u are given constants We shall build the solution u for any and take the

d d d e

limit as go es to zero This solution u is made of rarefaction or sho ckwaves propagating

c on the at b ottom areas that is either for x this waveis with the velo cities u



denoted L or for x this wave is denoted R On the slop e the remaining wave

will b e a stationary wave denoted byS

We get the form of a rarefaction waveby writing lo cally u as a function of h that

is u uh in the equations We get in the case of a at b ottom the linear

system



hu u h

t

 

hu u u huu gh h

x

Since h is exp ected to b e not constant the matrix is necessarily a singular matrix which

leads to the condition



hu g

This can b e seen as a dierential equation whose solutions are the so called Riemann

invariants

p

gh constant u



For a sho ckwave we use the Rankine Hugoniot relations and get that any state h u

can b e linked only to the states h u satisfying the jump condition

r

h h

g u u h h



hh

The o ccurrence of a sho ckwave or a rarefaction wave is ruled by the entropy condition

We get that a Lwaveischaracterized by

p

p

u gh gh for h h

g g g

q

u

hh

g

for h h u h h g

g g g

hh

g

ARwaveischaracterized by

ESAIM Proc Vol Septembre

Riemann Solvers for some Hyp erb olic Problems with a Source Term

p p

gh gh for h h u

d d d

q

u

hh

d

for h h u h h g

d d g

hh

d

It remains to consider the S wave We rst lo ok for a regular solution By writing

a a

d g

u uh as b efore we get the linear system where k g for x



hu u h

t

 

h hk hu u u huu gh

x

instead of The condition cannot b e satised for k This means that the

usual Riemann invariants play no role here Weintro duce a function hsuch that

 

h huh g

and solve Weget

d

 

huh h h k h h k

x t

dh

Hence h has the form

h k x C t

and since we get that

xt tx

d

huh

dh

that is

huh Ah B

where A and B are constants This allows us to have the expression of handofC t

in We get

B

h gh k x AtC

h

where C is a constant This expression means that a regular S wave is a solitary wave

and it is a stationary wave only for A that is for a constant ux hu In this case

u B

h gh which reads as a total energy Since u we get that is a convex

h

fonction of h going to innitywhenh or h and minimal for

B



h h

g

that is for

p





gh u u

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

which is the critical velo cityFrom we see that a regular stationary S wave has

these two prop erties

the mo de uvial or torrential can never change on the slop e

a critical velo city can b e reached only on the top of the slop e

the ux is constant that is A

Let us denote by h the value of h at x Wehave

l

B h u

l l

and from for x

h k h g a a h

l d g l

We see that the state at x do es not dep ends on Let us denote byh u this

r r

state and go to the limit as go es to zero whichchanges nothing in Weget

h u h u B

r r l l

h h g a a

r l d g

where the states h u and h u are of the same mo de We remark that the inuence

r r l l

of the prole of the slop e on the ow is limited to the value of the denivelation only

We consider now the o ccurrence of a stationary jump on the slop e and denote

by h and h the values of h on the right side and the left side of the sho ck resp ectively

From the Rankine Hugoniot conditions weget

h u h u B

h h

 

where h is a function such that h h h which gives here

B h

hg

h

We get from

h h

h h

The function hisconvex with a minimum at the critical p oint as for the function

h for the same ux B Since h h wehave necessarily that the states h u and

h u are of dierentmodes

For B the velo city eld is p ositive and from the entropy condition h h The

state h u is of uvial mo de and the state h u is of torrential mo de For B

the velo city eld is negative and from the entropy condition h h The state h u is

of torrential mo de and the state h u is of uvial mo de The p osition of the stationary

jump is determined by the initial data and can b e mo died when go es to zero Hence

the p osition of a stationary jump is unstable on the slop e and we prop ose to consider

only the cases of stationary jumps standing on the edges of the slop e That way the only

stationary jumps to b e considered will b e LorRwaves and not a S wave

ESAIM Proc Vol Septembre

Riemann Solvers for some Hyp erb olic Problems with a Source Term

The Riemann solver

The solution of the Riemann problem is the concatenation of L

S and Rwaves and we are concerned with the limit as go es to zero That way the

S wave will reduce to a jump obtained as the limit of a stationary wave of

p

gh instead of h since the Riemann We prop ose to work with the variables u and c

invariants b ecome linear in this case The initial data are

c u for x

g g

c u

c u for x

d d

and we set

K g a a

d g

We denote G c u and D c u and weintro duce the two curves GL and

g g d d

RD dened by

c foruu c

g g

u u c c for cc

g g g

GL

q

 

c c

cc

g g

for c c u u c c

g g g

cc

g

and

c for u u c

d d

u u c c for cc

d d d

RD

q





c c

cc

d

d

u u c c for c c

d d d

cc

d

Following GLasu go es from to makes c to go from to and following

RD asu go es from to makes c to go from to These two curves meet

in one p oint S c u which is unique for u c u c and c andu is

  d d g g  

undetermined in the other case

We denote by L c u the value of the solution on the left side of the S jump

l l



that is for x and by R c u the value on the right side that is for x

r r

These twopoints are linked bytwo conditions

L and R b elong to a constant ux curve

c u gQ Q constant f l ux

and Q is tted by the Bernoulli condition

h h K g a a

r l d g

which is a second linkage b etween L and R The condition also reads

c c

gQ

r

l

K c c

r l

c c

r

l

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

We rst consider the case K

Wehave several cases and wecho oze to present these cases according to the p osi

tion of S

If S corresp onds to a torrential mo de with u c wehave the following Case

 

C ase Either L G if u c G is torrential or L is the critical p ointu c

g g l l

of GLif u c SinceL is determined we can compute Q from and the p osition

g g

of R is obtained from with h h R is torrential That waywehave built L and

r l

R which is enough for a numerical solver We get the complete solution by solving a at

b ottom Riemann problem with c u as the left data and c u as the rightdataand

r r d d

taking the restriction to x of this solution

If S corresp onds to a uvial mo de we consider the critical p oint M c u of

M M

GL that is c u and the p oint N c u at the intersection of RD and the

M M N N

ux curve passing through M thatis

c u c u

N M

N M

Then we compute

K h h

max N M

Now wehavetwo p ossible cases

either K K and this case is similar to the Case above

max

or K K and wehave the following

max

C ase The p oint L will b elong to GL and the p oint R will b elong to RD

c u c u

r l

r l

we can compute b oth in uvial mo de Foragiven value of the ux Q

g g

the co ordinates of L and R and we can compute the quantity F Q h h

r l

which corresp onds to a functiontof QThus the problems reduces to a single equation of

the form

F Q K

which can b e solved by using the dichotomy pro cess for example

Wemay remark that for u with a negative ux the C ase still works When



the null ux is reached we get a ux with h a C onstant u from L to R for

h For h there is a wall reexion on the right side of the discontinuity and

l l

there is no ux

If S corresp onds to a torrential mo de with u c we consider the critical p oint

 

 

   

N c u onRD and the p oint M c u onGL b elonging to the same

N N M M

ux curve

 

c u c u

 

N M

N M

Then we compute

 

h K h

min

M N

ESAIM Proc Vol Septembre

Riemann Solvers for some Hyp erb olic Problems with a Source Term

Nowwehave three p ossible cases

either K K and this case is to b e solved as ab ovein Case

max

orK K K and this case is similar to the Case above

min max

orK K and we compute L and R as follows

min

C ase If G is torrential L and R are on the same ux curve that is c u c u

l r

r

l

L b elongs to the critical line u c and R b elongs to RD such that h h

l l r l

K h h This characterizes L and R and is sucientforanumerical computation

r l

The complete Riemann problem is solved by linking L to G by using the solution of a

usual at b ottom Riemann problem If G is uvial L is the critical p ointonGL and

we pro ceed as b efore to obtain R

If S corresp onds to a value c then the equation is trivialy satised



for any K and is undetermined The solution is the same as for a at b ottom

case since a dry ground app ears near the discontinuity of the top ography When the dry

ground app ears only on one side of the discontinuitywehaveawall reexion

For K we get a similar solver which is the same as ab ove for an observer

standing on the opp osite bank of the river We only have to use the new variables

u u u u h h h h K K

g d d g g d d g

instead of u u h h K in the solver describ ed ab ove

g d g d

A linearized Riemann solver

The Riemann solver describ ed ab oveworks in all cases and is rather dicult to handle

A rst linearized solver can b e obtained by using the Ro e technique with the variables

m hu and h Suchasolver may bring some values near the maxis which means a

small h and consequently a large velo city uFrom the CF L condition the time step will

severely fall down The previous analysis clearly shows that a linearization using c and u

may b e a b etter choice see also for the at b ottom case

We prop ose here an iterative metho d using the variables c and u where each step

is a linear pro cess For twostatesc u and c u we set

u u u c c c

u u u c c c

 

c c

 

h

g

Wehave

ucc g h u c u c u

and

u u

gh gh c c u u

ESAIM Proc Vol Septembre

Alain Yves LE ROUX

We start with twostatesc u and c u which can b e c u and c u and we

g g d d

c u and h as ab ove Then we iterate the following steps until convergence compute

compute c and u solution of the linear system

cuc g h u

c c u u K

compute c u cu solution of the linear system

u c u c

g g

u c u c

d d

c c c

u u u

either restart with c u c u orsetc c u u c c u u and stop

l l r r

u g h at least for K and always has Tosolve we need to have

a solution In practice no more than or iterations are needed And often only one is

enough which corresp onds to a really linearized solver

This solver works on a very wide class of cases but it is only an approximation

and the error may b ecome large when a strong sho ck o ccurs The prop erty of p ositiveness

of c c may b e lost to o

Conclusion

The Riemann solver including the source term allows to use the Go dunovscheme or any

nite volume scheme with a CFL condition which do es not dep end on the source term

It allows to work in a very wide class of values for the variables even in the linearized

version It can b e used in any dimension since we use it to compute a ux through an

interface which reduces to a one dimension problem This metho d has b een used for the

dimension simulation of a dam break on a random ground and is particularly ecient

near the obstacles or when recovering a dry ground or also when a dyr ground app ears

after the o o ding This solver may b e extended in order to include non linear friction

terms see Previous versions of this solver may b e found in or with many

numerical tests

References

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