Discontinuous Galerkin Method for Linear Wave Equations Involving Derivatives of the Dirac Delta Distribution

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