论文标题
线性波方程的不连续的Galerkin方法，涉及Dirac Delta分布的衍生物
Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution
论文作者
论文摘要
线性波方程由Dirac Delta分布$δ（X）$及其导数（S）来源，可以作为许多不同现象的模型。我们描述了一种不连续的Galerkin（DG）方法，以数字求解与$ \ partial^nδ/\ partial x^n $成正比的源项。尽管存在奇异的源术语，这意味着不连续或潜在的奇异解决方案，但我们的DG方法即使在源位置也达到了全球光谱精度。我们的DG方法是为以完全一阶形式编写的波方程开发的。使用分布辅助变量进行一阶还原，该变量消除了一些源项的单数行为。尽管这在数值上很有帮助，但会产生分布约束。我们表明，如果初始约束违规与$δ（x）$成正比，则可以开发与时间无关的伪造解决方案。数值实验通过与精确的解决方案进行比较来验证该行为和我们方案的收敛性。
Linear wave equations sourced by a Dirac delta distribution $δ(x)$ 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 $\partial^n δ/\partial x^n$. 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 $δ(x)$. Numerical experiments verify this behavior and our scheme's convergence properties by comparing against exact solutions.
