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.

翻译：由狄拉克δ分布$\delta(x)$及其导数激发的线性波动方程可模拟多种物理现象。本文提出一种间断伽辽金（DG）方法，用于数值求解含有$\partial^n \delta /\partial x^n$比例源项的此类方程。尽管奇异源项会导致不连续甚至奇异解的存在，我们的DG方法即使在源点位置也能实现全局谱精度。该方法基于完全一阶形式的波动方程开发，通过引入分布型辅助变量进行降阶处理，消去部分源项的奇异性。虽然这种处理在数值上具有优势，但会引入分布约束条件。研究表明，若初始约束违背量与$\delta(x)$成正比，则可能产生与时间无关的虚假解。数值实验通过对比精确解验证了该现象及方案的收敛特性。