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Gosse, Laurent

A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. (English) Zbl 1018.65108

Math. Models Methods Appl. Sci. 11, No. 2, 339-365 (2001).
Summary: The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a non-conservative reformulation of the zero-order terms of the right-hand side of the equations. In this context, we decided to use the results of G. Dal Maso, P. G. Le Floch and F. Murat about non-conservative products [J. Math. Pures Appl. IX. Sér. 74, No. 6, 483-548 (1995; Zbl 0853.35068)], and the generalized Roe matrices introduced by I. Toumi [J. Comput. Phys. 102, No. 2, 360-373 (1992; Zbl 0783.65068)] to derive a first-order linearized well-balanced scheme in the sense of J. M. Greenberg and A.-Y. Le Roux [SIAM J. Numer. Anal. 33, No. 1, 1-16 (1996; Zbl 0876.65064)]. As a main feature, this approach is able to preserve the right asymptotic behavior of the original inhomogeneous system, which is not an obvious property. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations.

Cited in 66 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76N15 Gas dynamics (general theory)

Keywords:

convergence; Cauchy problem; numerical results; hyperbolic systems of conservation laws with source terms; generalized Roe matrices; Euler equations

Citations:

Zbl 0853.35068; Zbl 0783.65068; Zbl 0876.65064
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