Consistency of high-order Godunov schemes

A bstract: 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 arc Lipschitz continuous with respect- to the spatial coordinat-cs.lt- 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 dynamicist-s. Since high-resolution schemes require the computation of several quantities per mesh cell (c.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].