Sharp Penalty Term and Time Step Bounds for the Interior Penalty Discontinuous Galerkin Method for Linear Hyperbolic Problems

Abstract

We present sharp and sufficient bounds for the interior penalty term and time step size to ensure stability of the Symmetric Interior Penalty Discontinuous Galerkin (SIPDG) method combined with an explicit time-stepping scheme. These conditions hold for generic meshes, including unstructured non-conforming heterogeneous meshes of mixed element types, and apply to a large class of linear hyperbolic problems, including the acoustic wave equation, the (an)isotropic elastic wave equations and Maxwell's equations. The penalty term bounds are computed element-wise, while bounds for the time step size are computed at weighted submeshes requiring only a small number of elements and faces. Numerical results illustrate the sharpness of these bounds.