Consistency of high-order Godunov schemes

This paper focuses on a rigorous treatment of consistency of high order numerical fluxes in dealing with nonlinear hyperbolic conservation laws. It demonstrates how a clear understanding of consistency leads to stability and convergence . The classical paper [18] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Its novelty was the suggestion to use local analytic solutions in the construction of numerical fluxes. The seminal paper of van Leer [32] 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 [2] provided analytical solutions (for piecewise linear data) that could be readily implemented in a high-resolution robust code. The first significant observation made here is that under very mild conditions the associated fluxes are Lipschitz continuous with respect to the spatial coordinates.It entails the result that a weak solution is indeed a solution to the corresponding balance law (obtained by a formal application of the Gauss-Green formula), thus closing the gap between the mathematical notion of a “weak solution” and the formalism of “balance law” (integral formulation) favored by fluid dynamicists. 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. Mathematically, it is the adaption of the Lax-Wendroff methodology in the setting of discontinuous solutions. A combination of the appropriate consistency hypothesis and stability of the scheme leads to a convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [20].