Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution

TL;DR

This paper develops a discontinuous Galerkin method for linear wave equations with singular source terms involving derivatives of the Dirac delta, achieving spectral accuracy despite the singularities.

Contribution

The paper introduces a DG scheme that handles singular delta source terms in wave equations with spectral accuracy, using a distributional auxiliary variable and addressing constraint violations.

Findings

The DG method attains global spectral accuracy at the source location.

Numerical experiments confirm convergence and behavior of the scheme.

A spurious solution can develop if initial constraints violate the delta distribution.

Abstract

Linear wave equations sourced by a Dirac delta distribution $\delta(x)$ and its derivative(s) can serve as a model for many different phenomena. We describe a discontinuous Galerkin (DG) method to numerically solve such equations with source terms proportional to $\partial^n \delta /\partial x^n$. Despite the presence of singular source terms, which imply discontinuous or potentially singular solutions, our DG method achieves global spectral accuracy even at the source's location. Our DG method is developed for the wave equation written in fully first-order form. The first-order reduction is carried out using a distributional auxiliary variable that removes some of the source term's singular behavior. While this is helpful numerically, it gives rise to a distributional constraint. We show that a time-independent spurious solution can develop if the initial constraint violation is proportional to $\delta(x)$. Numerical experiments verify this behavior and our scheme's convergence properties by comparing against exact solutions.