A Godunov-Type Solver for the Numerical Approximation of Gravitational Flows
J. Vides 1*, B. Braconnier 2, E. Audit 3, C. Berthon 4, B. Nkonga 51 Inria, Maison de la Simulation, USR 3441, Gif-sur-Yvette, France.
2 IFP Energies Nouvelles, 1-4 avenue de Bois-Preau, 92852 Rueil-Malmaison, France.
3 CEA, Maison de la Simulation, USR 3441, Gif-sur-Yvette, France.
4 Univ. of Nantes, Lab. J. Leray, UMR CNRS 6629, Nantes, France.
5 Univ. of Nice-Sophia Antipolis, Lab. J.A. Dieudonne, UMR CNRS 7351, Nice, France; Inria Sophia Antipolis, BP. 93, F-06902 Sophia Antipolis Cedex, France.
Received 6 July 2012; Accepted (in revised version) 21 March 2013
Available online 26 July 2013
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  and several numerical experiments involving gravitational flows for astrophysics highlight the scheme.AMS subject classifications: 85-08, 65Z05, 35L65, 35L45, 76M12
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Key words: Euler-Poisson equations, approximate Riemann solver, relaxation scheme, source terms, gravitational effects.
Email: firstname.lastname@example.org (J. Vides), email@example.com (B. Braconnier), firstname.lastname@example.org (E. Audit), email@example.com (C. Berthon), firstname.lastname@example.org (B. Nkonga)