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

## TL;DR

This paper derives sharp bounds for the penalty term and time step size in the SIPDG method to ensure stability across various linear hyperbolic problems and complex mesh configurations.

## Contribution

It provides the first sharp, element-wise bounds for penalty and time step size applicable to a wide range of hyperbolic PDEs on unstructured meshes.

## Key findings

- Bounds are sharp and numerically validated.
- Applicable to diverse hyperbolic equations and mesh types.
- Element-wise computation of bounds enhances practical implementation.

## 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.