A finite difference - discontinuous Galerkin method for the wave equation in second order form

TL;DR

This paper introduces a hybrid finite difference-discontinuous Galerkin method for the wave equation that combines efficiency and flexibility, achieving high-order accuracy and stability through a novel coupling and error analysis.

Contribution

The paper presents a new hybrid discretization method coupling finite difference and discontinuous Galerkin techniques with a penalty interface, ensuring stability and optimal convergence for the wave equation.

Findings

Achieves fourth order convergence with specific method combination

Ensures stability via a penalty coupling technique

Validated through numerical experiments

Abstract

We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility of discontinuous Galerkin methods on unstructured meshes. The two spatial discretizations are coupled by a penalty technique at the interface such that the overall semidiscretization satisfies a discrete energy estimate to ensure stability. In addition, optimal convergence is obtained in the sense that when combining a fourth order finite difference method with a discontinuous Galerkin method using third order local polynomials, the overall convergence rate is fourth order. Furthermore, we use a novel approach to derive an error estimate for the semidiscretization by combining the energy method and the normal mode analysis for a corresponding one dimensional model problem. The stability and accuracy analysis are verified in numerical experiments.