Robust and Accurate Approximation of Hyperbolic Systems

Abstract

The project consists of developing robust numerical methods for solving hyperbolic systems of conservation laws such as the Euler equations, the shallow water equations, and related systems such as radiative hydrodynamics. Robustness is an area of investigation that is not well addressed in the literature. We think that most current high-order methods are not robust. This lack of robustness is what makes them unattractive to practitioners. Real progresses will come when robust accurate methods are readily available to a large community of practitioners. We do not think that this is the case at the moment. Our research program is articulated around the following five points: (i) The numerical method must be invariant domain preserving (i.e. preserve convex invariant domains of the problem) on any unstructured meshes in any space dimension; (ii) The method must be robust in the sense that it should not involve any tuning parameter, mesh-dependent coefficient, or problemdependent stabilization. A robust method should also be easy to program and parallelize. It should not require any subtle mathematical knowledge from practitioners to run it; (iii) The method must be at least third-order accurate in space and time and must be open to higher-order extensions; (iv) The method should respect the physical dissipation of the PDE; (v) The above objectives must be reached by stating precise statements supported either by mathematical proofs or very strong numerical evidences