Convergence of a Finite Volume Scheme for Compactly Heterogeneous Scalar Conservation Laws

TL;DR

This paper develops a finite volume scheme for scalar conservation laws with heterogeneous flux functions, proving convergence to the entropy solution using discontinuous flux theory.

Contribution

It introduces a novel finite volume scheme tailored for scalar conservation laws with convex, heterogeneous flux functions and proves its convergence to the entropy solution.

Findings

The scheme converges boundedly almost everywhere.

Convergence is established for a wide class of flux functions.

The approach leverages the theory of discontinuous flux.

Abstract

We build a finite volume scheme for the scalar conservation law $\partial_t u + \partial_x (H(x, u)) = 0$ with bounded initial condition for a wide class of flux function $H$, convex with respect to the second variable. The main idea for the construction of the scheme is to use the theory of discontinuous flux. We prove that the resulting approximating sequence converges boundedly almost everywhere on $\mathopen]0, +\infty\mathclose[$ to the entropy solution.