Role of Fluxes in High-Order Godunov Schemes

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 [15] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. The seminal paper of van Leer [30] 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 [1] 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 [17].