An energy-based summation-by-parts finite difference method for the wave equation in second order form

TL;DR

This paper introduces an energy-based finite difference method for the wave equation in second order form that ensures stability and accuracy without mesh-dependent penalty parameters, verified through numerical experiments.

Contribution

The paper presents a novel energy-based SBP finite difference method that simplifies boundary and interface condition enforcement without penalty parameters.

Findings

Method achieves stable energy estimates.

Numerical experiments confirm convergence and robustness.

No mesh-dependent penalty parameters needed.

Abstract

We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method.