Moving Mesh Discontinuous Galerkin Methods for PDEs with Traveling Waves

TL;DR

This paper introduces a moving mesh discontinuous Galerkin method tailored for nonlinear PDEs with traveling wave solutions, effectively capturing sharp wave fronts and speeds with high accuracy.

Contribution

The paper develops a novel moving mesh dG method that decouples PDE solution from mesh movement, improving resolution of traveling waves in nonlinear PDEs.

Findings

Accurately resolves sharp wave fronts and speeds.

Efficiently solves Burgers', Burgers'-Fisher, and Schl"ogl equations.

Demonstrates high accuracy and efficiency of the method.

Abstract

In this paper, a moving mesh discontinuous Galerkin (dG) method is developed for nonlinear partial differential equations (PDEs) with traveling wave solutions. The moving mesh strategy for one dimensional PDEs is based on the rezoning approach which decouples the solution of the PDE from the moving mesh equation. We show that the dG moving mesh method is able to resolve sharp wave fronts and wave speeds accurately for the optimal, arc-length and curvature monitor functions. Numerical results reveal the efficiency of the proposed moving mesh dG method for solving Burgers', Burgers'-Fisher and Schl\"ogl(Nagumo) equations.