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A Godunov-Type Solver for the Numerical Approximation of Gravitational Flows

Published online by Cambridge University Press:  03 June 2015

J. Vides ,
B. Braconnier ,
E. Audit ,
C. Berthon  and
B. Nkonga
J. Vides*
Affiliation:
Inria, Maison de la Simulation, USR 3441, Gif-sur-Yvette, France
B. Braconnier*
Affiliation:
IFP Energies Nouvelles, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
E. Audit*
Affiliation:
CEA, Maison de la Simulation, USR 3441, Gif-sur-Yvette, France
C. Berthon*
Affiliation:
Univ. of Nantes, Lab. J. Leray, UMR CNRS 6629, Nantes, France
B. Nkonga*
Affiliation:
Univ. of Nice-Sophia Antipolis, Lab. J.A. Dieudonné, UMR CNRS 7351, Nice, France Inria Sophia Antipolis, BP. 93, F-06902 Sophia Antipolis Cedex, France
*
Corresponding author.Email:jeaniffer-lissette.vides_higueros@inria.fr
Email:benjamin.braconnier@ifpenergiesnouvelles.fr
Email:edouard.audit@cea.fr
Email:christophe.berthon@math.univ-nantes.fr
Email:boniface.nkonga@unice.fr
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Abstract

We present a new numerical method to approximate the solutions of an Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity source term involved in the latter equations. In order to approximate this source term, its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The method has been implemented in the software HERACLES [29] and several numerical experiments involving gravitational flows for astrophysics highlight the scheme.

Keywords

85-0865Z0535L6535L4576M12Euler-Poisson equationsapproximate Riemann solverrelaxation schemesource termsgravitational effects
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 1 , January 2014 , pp. 46 - 75
Copyright
Copyright © Global Science Press Limited 2014

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References

[1] Abgrall, R., Karni, S., A relaxation scheme for the two-layer shallow water system, Hyperbolic Problems: Theory, Numerics, Application, Benzoni-Gavage, S. and Serre, D. Editors, Springer, 135–144, 2008.Google Scholar
[2] Aregba-Driollet, D., Berthon, C., Numerical approximation of Kerr-Debye equations, preprint hal-00293728.Google Scholar
[3] Balbus, S., Papaloizou, J.C.B., On the dynamical foundations of α disks, Astrophysical Journal 521 (1999), p. 650.CrossRefGoogle Scholar
[4] Baudin, M., Berthon, C., Cocquel, F., Masson, R., Tran, Q.H., A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math. 99 (2005), 411–440.CrossRefGoogle Scholar
[5] Berthon, C., Stability of the MUSCL schemes for the Euler equations, Comm. Meth. Sci. 3 (2005), 133–158.Google Scholar
[6] Berthon, C., Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys. 218 (2006), 495–509.CrossRefGoogle Scholar
[7] Berthon, C., Numerical approximations of the 10-moment Gaussian closure, Math. Comput. 75 (2006), 1809–1831.CrossRefGoogle Scholar
[8] Berthon, C., Braconnier, B., Nkonga, B., Numerical approximation of a degenerated nonconservative multifluid model: relaxation scheme, International Journal for Numerical methods in Fluids 48 (2005), 85–90.CrossRefGoogle Scholar
[9] Berthon, C., Breuss, M., Titeux, M.O., A relaxation scheme for pressureless gas dynamics, Num. Method Partial Diff. Equations 22 (2006), 484–505.Google Scholar
[10] Berthon, C., Charrier, P., Dubroca, B., An HLLC scheme to solve the M1 Model of radiative transfer in two space dimensions, J. Sci. Comput. 31 (2007), 347–389.CrossRefGoogle Scholar
[11] Berthon, C., Marche, F., A positive preserving hight order VFRoe scheme for shallow water equations: A class of relaxation schemes, SIAM J. Sci. Comput. 30 (2008), 2587–2612.CrossRefGoogle Scholar
[12] Bouchut, F., Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), 623–672.CrossRefGoogle Scholar
[13] Bouchut, F., Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhauser, 2004.CrossRefGoogle Scholar
[14] Bouchut, F., Morales, T., An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, M2AN 42 (2008), 683–698.CrossRefGoogle Scholar
[15] Chalons, C., Coquel, F., Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, Numer. Math. 101 (2005), 451–478.CrossRefGoogle Scholar
[16] Chalons, C., Coulombel, J.-F., Relaxation approximation of the Euler equations, J. Math. Anal. Appl. 348 (2008), 872–893.CrossRefGoogle Scholar
[17] Chen, G.Q., Levermore, C.D., Liu, T.P., Hyperbolic conservation laws with stiff relaxations terms and entropy, Comm. Pure Appl. Math. 47 (1995), 787–830.Google Scholar
[18] Chièze, J.-P., Elements of Hydrodynamics Applied to the Interstellar Medium, Starbursts: Triggers, Nature, and Evolution, Les Houches School, September 17-27, 1996. ESP Sciences 1998, p. 77, Editors Guiderdoni, Bruno and Kembhavi, Ajit.Google Scholar
[19] Coquel, F., Perthame, B., Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal 35 (1998), 2223–2249.CrossRefGoogle Scholar
[20] Crispel, P., Degond, P., Vignal, M.-H., An asymptotically stable discretization for the Euler-Poisson system in the quasineutral limit, C.R. Acad. Sci. Paris Ser. I (2005), 323–328.Google Scholar
[21] Crispel, P., Degond, P., Vignal, M.-H., An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phys. 223 (2007), 208–234.CrossRefGoogle Scholar
[22] Degond, P., Deluzet, F., Sangam, A., Vignal, M.-H., An Asymptotic Preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phys. 228 (2009), 3540–3558.CrossRefGoogle Scholar
[23] Degond, P., Jin, S., Liu, J.-G., Mach-number uniform asymptotic-preserving Gauge schemes for compressible flows, Bull. Inst. Math., Acad. Sinica (New Series) 2 (2007), 851–892.Google Scholar
[24] Degond, P., Liu, J.-G., Vignal, M.-H., Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit, SIAM J. Numer. Anal. 46 (2008), 1298–1322.CrossRefGoogle Scholar
[25] Fabre, S., Stability analysis of the Euler-Poisson equations, J. Comput. Phys. 101 (1992), 445–451.CrossRefGoogle Scholar
[26] Gallouet, T., Hérard, J. M., Seguin, N., Some approximate Godunov schemes to compute shallow-water equations with topography, Computers and Fluids 32 (2003), 479–513.CrossRefGoogle Scholar
[27] Gallardo, J.M., Parés, C., Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys. 227 (2007), 574–601.CrossRefGoogle Scholar
[28] Godlewski, E., Raviart, P.A., Hyperbolic systems of conservations laws, Applied Mathematical Sciences, 118, Springer, 1996.CrossRefGoogle Scholar
[29] Gonzáles, M., Audit, E., Huynh, P., HERACLES: a three-dimensional radiation hydrodynamics code, Astronomy Astrophysics 464 (2007), 429–435.Google Scholar
[30] Greenberg, J.M., Leroux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), 1–16.CrossRefGoogle Scholar
[31] Greenberg, J.M., Leroux, A.Y., Baraille, R., Noussair, A., Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal. 34 (1997), 1980–2007.CrossRefGoogle Scholar
[32] Harten, A., Lax, P.D., Leer, B.Van, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM 25 (1983), 33–61.CrossRefGoogle Scholar
[33] James, R.A., The solution of Poisson’s equation for isolated source distributions, J. Comput. Phys. 25 (1977), 71–93.CrossRefGoogle Scholar
[34] Jin, S., A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN Math. Model. Numer. Anal. 35 (2001), 631–645.CrossRefGoogle Scholar
[35] Jin, S., Xin, Z.P., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235–278.CrossRefGoogle Scholar
[36] Kippenhahn, R., Weigert, A., Stellar Structure and Evolution, Springer, Berlin (a.o.), 1990.CrossRefGoogle Scholar
[37] LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys. 146 (1998), 346–365.CrossRefGoogle Scholar
[38] LeVeque, R.J., Pelanti, M., A class of approximate Riemann solvers and their relation to relaxation schemes, J. Comput. Phys. 172 (2001), 572–591.CrossRefGoogle Scholar
[39] Li, Y., Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations, J. Math. Anal. Appl. 342 (2008), 1107–1125.CrossRefGoogle Scholar
[40] Natalini, R., Rousset, F., Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations, Proc. Amer. Math. Soc. 134 (2006), 2251–2258.CrossRefGoogle Scholar
[41] Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys. 213 (2006), 474–499.CrossRefGoogle Scholar
[42] Roe, P.L., Characteristic-based schemes for the Euler equations, Ann. Rev. Fluid Mech. 18 (1986), 337–365.CrossRefGoogle Scholar
[43] Suliciu, I., On modelling phase transitions by means of rate-type constitutive equations, shock wave structure, Int. J. Engrg. Sci. 28 (1990), 829–841.CrossRefGoogle Scholar
[44] Suliciu, I., Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations, Int. J. Engrg. Sci. 30 (1992), 483–494.CrossRefGoogle Scholar
[45] Suresh, A., Positivity-preserving schemes in multidimensions, SIAM Sci. Comput. 22 (2000), 1184–1198.CrossRefGoogle Scholar
[46] Leer, B.van, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme, J. Comput. Phys. 14 (1974), 361–70.Google Scholar
[47] Leer, B. van, Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method, J. Comput. Phys. 32 (1979), 101–136.Google Scholar
[48] Leer, B. van, On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Statist. Comput. 5 (1984), 1–20.Google Scholar
[49] Weyl, H., Shock waves in arbitrary fluids, Comm. Pure Appl. Math. 2 (1949), 103–122.CrossRefGoogle Scholar
[50] Whitham, J., Linear and Nonlinear Waves, Wiley, New-York, 1974.Google Scholar