Convergence of discontinuous Galerkin schemes for front propagation with obstacles

TL;DR

This paper analyzes the convergence of semi-Lagrangian and Runge-Kutta discontinuous Galerkin schemes for nonlinear Hamilton-Jacobi equations modeling front propagation with obstacles, providing new error bounds under minimal regularity assumptions.

Contribution

It introduces new convergence results and error bounds for DG schemes applied to obstacle-involved Hamilton-Jacobi equations with low regularity data.

Findings

Established convergence of SLDG and RKDG schemes.

Derived error bounds for Lipschitz continuous data.

Validated results for equations with natural low regularity.

Abstract

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These "low regularity" assumptions are the natural ones for the solutions of the studied equations.