Stable and accurate hybrid finite volume methods based on pure convexity arguments for hyperbolic systems of conservation law
Abstract
This exploratory work tries to present first results of a novel approach for the numerical approximation of solutions of hyperbolic systems of conservation laws. The objective is to define stable and "reasonably" accurate numerical schemes 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 twophase models where solutions of Riemann problems are hard, see impossible to compute. For Riemann or Roelike solvers, each fluid 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 Rdiagonalizable 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 secondorder LaxWendroff scheme flux and the firstorder modified LaxFriedrichs scheme flux with an "optimal" combination rate that ensures both minimal numerical dissipation and good accuracy. The resulting scheme is a central schemelike method. We will also need and propose a definition of local dissipation by convexity for hyperbolic or elliptichyperbolic systems. This convexity argument allows us to overcome the difficulty of nonexistence of classical entropyflux pairs for certain systems. We emphasize the systematic feature of the method which can be fastly implemented or adapted to any kind of systems, with general analytical or datatabulated equations of state. The numerical results presented in the paper are not superior to many existing stateoftheart numerical methods for conservation laws such as ENO, MUSCL or central scheme of Tadmor and coworkers. The interest is rather the systematic feature of the method and its very fast implementation for prototypying and fluid model validation. In this context, the Rusanov scheme is often used; the present approach here gives far better results.
 Publication:

Journal of Computational Physics
 Pub Date:
 January 2004
 DOI:
 10.1016/j.jcp.2003.07.028
 Bibcode:
 2004JCoPh.193..426D