A simple numerical scheme for the 3-D system of ideal gases and a study of approximation in the sense of distributions

• Autores: M. Colombeau
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 248, Nº 1, 2013, págs. 15-30
• Idioma: inglés
• DOI: 10.1016/j.cam.2013.01.001
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• Resumen

  • We present a 3-D numerical scheme for the approximation of the system of gas dynamics.

    We prove that, as long as the boundedness of the velocity field in the CFL condition and the positiveness of the total energy are numerically verified when the space step tends to 0, the scheme provides a numerical solution which satisfies the conservation laws in the sense of distributions modulo a small remainder of order one in the space step (and a slightly weaker order for the state law in the case of irregular approximate solutions). In the case of classical tests (Sod, Woodward�Colella, Toro) one provides numerical verifications that these conditions are verified even for very small values of the space step and correspondingly one observes that the scheme produces the known exact solutions with great precision. Full proofs are given in the pressureless case. The scheme produces the numerical results on the six 2-D Riemann problems presented by P.D. Lax [P.D. Lax, Mathematics and Physics, Bull. AMS 45 (1) (2008) 135�152], up to the smallest detail. Therefore this result contributes to a mathematical justification of the fact that the numerical results presented by P.D. Lax on these 2-D Riemann problems are approximations of a solution. This simple order one low-cost 3-D scheme is obtained from the convection-pressure correction method proposed by Le Roux et al. [R. Baraille, G. Bourdin, F. Dubois, A.Y. Le Roux, Une version à pas fractionnaire du schèma de Godunov pour l�hydrodynamique, Comptes Rendus Acad. Sci. Paris 314 (1992) 147�152].