Convergence of a Godunov scheme for degenerate conservation laws with BV spatial flux and a study of Panov type fluxes

TL;DR

This paper proves the convergence of the Godunov scheme for scalar conservation laws with complex, discontinuous, and degenerate flux functions, extending previous results to more general flux scenarios with accumulation points and degeneracies.

Contribution

It extends convergence proofs of the Godunov scheme to flux functions with multiple discontinuities, degeneracies, and non-invertible singular maps, using new proof techniques.

Findings

Convergence of Godunov scheme for complex flux functions.

New proof methods for degenerate and Panov type fluxes.

Numerical examples illustrating theoretical results.

Abstract

In this article we prove convergence of the Godunov scheme of [16] for a scalar conservation law in one space dimension with a spatially discontinuous flux. There may be infinitely many flux discontinuities, and the set of discontinuities may have accumulation points. Thus the existence of traces cannot be assumed. In contrast to the study appearing in [16], we do not restrict the flux to be unimodal. We allow for the case where the flux has degeneracies, i.e., the flux may vanish on some interval of state space. Since the flux is allowed to be degenerate, the corresponding singular map may not be invertible, and thus the convergence proof appearing in [16] does not pertain. We prove that the Godunov approximations nevertheless do converge in the presence of flux degeneracy, using an alternative method of proof. We additionally consider the case where the flux has the form described in [21]. For this case we prove convergence via yet another method. This method of proof provides a spatial variation bound on the solutions, which is of independent interest. We present numerical examples that illustrate the theory.