An energy-based discontinuous Galerkin method for the wave equation with nonsmooth solutions

TL;DR

This paper introduces a stable, high-order discontinuous Galerkin method for the wave equation that effectively handles nonsmooth solutions and suppresses spurious oscillations, validated through stability analysis and numerical experiments.

Contribution

It combines energy-based DG methods with oscillation-free techniques to improve accuracy and robustness for nonsmooth wave solutions, including nonlinear cases.

Findings

Achieves optimal convergence rates for smooth solutions

Maintains oscillation-free behavior for nonsmooth solutions

Proven stability and error estimates

Abstract

We develop a stable and high-order accurate discontinuous Galerkin method for the second order wave equation, specifically designed to handle nonsmooth solutions. Our approach integrates the energy-based discontinuous Galerkin method with the oscillation-free technique to effectively suppress spurious oscillations near solution discontinuities. Both stability analysis and apriori error estimates are established for common choices of numerical fluxes. We present a series of numerical experiments to confirm the optimal convergence rates for smooth solutions and its robustness in maintaining oscillation-free behavior for nonsmooth solutions in wave equations without or with nonlinear source terms.