Preprint 2003-012

Stable and Accurate Hybrid Finite Volume Methods Based on Pure Convexity Arguments for Hyperbolic Systems of Conservation Laws

Florian De Vuyst

Abstract: The objective of this work is to define stable and accurate numerical schemes for the approximation of hyperbolic systems of conservation laws while being free from any upwind process and from any computation of derivatives or mean Jacobian matrices. That means that we only want to perform flux evaluations. This would be useful for ``complicated'' systems like those of two-phase models where solutions of Riemann problems are hard to compute, see unreachable. More, each model needs the particular computation of the Jacobian matrix of the flux and the hyperbolicity property which can be conditional for some of these models makes the matrices be not $\R$-diagonalizable everywhere in the admissible state space. In this paper, we rather propose some numerical schemes where the stability is obtained using convexity considerations. A certain rate of accuracy is also expected. For that, we propose to build numerical hybrid fluxes that are convex combinations of the second order Lax-wendroff scheme flux and the first order modified Lax-Friedrichs scheme flux with an ``optimal'' combination rate that ensures both minimal numerical dissipation and optimal accuracy (in a certain sense). We will also need and propose a definition of local dissipation by convexity rate for hyperbolic or elliptic-hyperbolic systems. This convexity argument allows us to overcome the difficulty of non existence of classical entropy-flux pairs for certain systems. We emphasize the generic feature of the method which can be fastly implemented or adapted to any kind of systems, with general analytical or data-tabulated equations of state.

Available as PDF (2.1 Mbytes).
Florian De Vuyst, <>
Publishing information:
Revised version received July 15 2003.
Source codes of the numerical experiments written in Matlab can be downloaded and freely used from the authors webpage.
Submitted by:
<> January 31 2003.

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