Moving mesh discontinuous Galerkin methods for PDEs with traveling waves

• Moving mesh with discontinuous Galerkin method.
• Nonlinear 1D problems with traveling waves.
• Resolving sharp moving fronts and determination of correct wave speed.
• Uncoupling of the discretization and mesh equations.

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 Schlögl (Nagumo) equations.