The discontinuous Galerkin method for fractal conservation laws

TL;DR

This paper develops and analyzes a discontinuous Galerkin method tailored for fractal conservation laws, providing stability, error estimates, and convergence rates, supported by numerical experiments for linear and nonlinear cases.

Contribution

It introduces a novel DG method specifically designed for fractal conservation laws, with rigorous stability, error, and convergence analyses.

Findings

Established stability estimates for the method.

Proved convergence rates toward entropy solutions.

Validated effectiveness through numerical experiments.

Abstract

We propose, analyze, and demonstrate a discontinuous Galerkin method for fractal conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and whenever piecewise constant elements are utilized, we prove a rate of convergence toward the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractal conservation laws.