A Godunov type scheme and error estimates for multidimensional scalar conservation laws with Panov-type discontinuous flux

TL;DR

This paper introduces a Godunov-type scheme for multidimensional scalar conservation laws with discontinuous flux, proving convergence, uniqueness, and optimal error estimates, and providing numerical validation.

Contribution

It develops the first multidimensional Godunov scheme with proven convergence and error estimates for conservation laws with discontinuous flux.

Findings

Proved uniqueness of entropy solutions.

Established convergence of the scheme at an optimal rate.

Validated the theory with numerical examples.

Abstract

This article concerns a scalar multidimensional conservation law where the flux is of Panov type and may contain spatial discontinuities. We define a notion of entropy solution and prove that entropy solutions are unique. We propose a Godunov-type finite volume scheme and prove that the Godunov approximations converge to an entropy solution, thus establishing existence of entropy solutions. We also show that our numerical scheme converges at an optimal rate of $\mathcal{O}(\sqrt{\D t}).$ To the best of our knowledge, convergence of the Godunov type methods in multi-dimension and error estimates of the numerical scheme in one as well as in several dimensions are the first of it's kind for conservation laws with discontinuous flux. We present numerical examples that illustrate the theory.