A Non-stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces

TL;DR

This paper introduces a stable, non-stiff finite difference method for the scalar wave equation that effectively handles characteristic boundary conditions and nonlinear interfaces using an upwind approach with characteristic variables.

Contribution

The authors develop a novel technique that avoids stiffness in summation-by-parts finite difference methods by employing characteristic variables and an additional boundary variable.

Findings

The scheme is energy stable for the scalar anisotropic wave equation.

Numerical experiments confirm accuracy and robustness.

The method's time step restriction depends on wave propagation, not boundary closure.

Abstract

Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to "upwind" the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.