Role of Fluxes in High-Order Godunov Schemes

Abstract

This paper focuses on the evolution of some mathematical aspects related to high-resolution approximations to nonlinear hyperbolic balance laws. It addresses the crucial role of numerical fluxes in dealing with the three concepts of consistency, stability and convergence. The classical paper by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. The seminal paper of van Leer has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The fundamental step consists of modifying the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise polynomial (rather than piecewise constant) initial data. The GRP (Generalized Riemann Problem) analysis provided analytical solutions (for piecewise linear data) that could be readily implemented in a high-resolution robust code. The treatment utilizes the framework of balance laws, a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula), and the associated fluxes are Lipschitz continuous with respect to the spatial coordinates. Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of flux consistency must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff.