Speaker: Chi-Wang Shu, Brown University
Title: Discontinuous Galerkin method for hyperbolic equations with delta-singularities
Abstract: Discontinuous Galerkin (DG) methods are finite element methods with features from high resolu- tion finite difference and finite volume methodologies and are suitable for solving hyperbolic equations with nonsmooth solutions. In this talk we will describe our recent work on the study of DG methods for solving hyperbolic equations with $\delta$-singularities in the initial condition, in the source term, or in the solutions. For such singular solutions, many numerical techniques rely on modifications with smooth kernels and hence may severely smear such singularities, leading to large errors in the approximation. On the other hand, the DG methods are based on weak formula- tions and can be designed directly to solve such problems without modifications, leading to very accurate results. We will discuss both error estimates for model linear equations, in negative norm and in strong norm after post-processing, and applications to nonlinear systems including the rendez-vous systems and pressureless Euler equations involving $\delta$-singularities in their solutions. For the nonlinear case a high order accuracy bound-preserving limiter is crucial to maintain nonlinear
stability and to avoid blowups of the numerical solution. This is joint work with Yang Yang, Dongming Wei and Xiangxiong Zhang.
