Colloquium

Discontinuous Galerkin (DG) methods are finite element methods with features from high resolution finite difference and finite volume methodologies and are suitable for solving hyperbolic equations with nonsmooth solutions. In this talk we will first give a survey on DG methods, then we will describe our recent work on the study of DG methods for solving hyperbolic equations with singularities in the initial condition, in the source term, or in the solutions. The type of singularities include both discontinuities and \(\delta\)-functions. Especially for problems involving \(\delta\)-singularities, 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 formulations 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 and applications to nonlinear systems including the rendez-vous systems and pressureless Euler equations involving \(\delta\)-singularities in their solutions. This is joint work with Qiang Zhang, Yang Yang and Dongming Wei.