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Riemann Solvers for Some Hyperbolic Problems with a Source Term

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Riemann Solvers for Some Hyperbolic Problems with a Source Term 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 maintenir alequilibre 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 Riemann 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 matrix 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 Riemann solver 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
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