Numerical treatment of interfaces for second-order wave equations

TL;DR

This paper introduces a novel numerical scheme for second-order wave equations that efficiently handles interfaces between grids, reducing data exchange while maintaining accuracy, crucial for parallel multi-dimensional simulations.

Contribution

It develops a penalty-type interface scheme for second-order wave equations that requires less data transfer than existing methods, enhancing parallel computation efficiency.

Findings

The scheme preserves the norm of the solution.

It uses standard finite-difference operators with summation by parts.

A semi-implicit IMEX Runge-Kutta method is employed for time integration.

Abstract

In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving the second-order wave equation. We show that it is possible to implement an interface scheme of "penalty" type for the second-order wave equation, similar to the ones used for first-order hyperbolic and parabolic equations, and the second-order scheme used by Mattsson et al (2008). These schemes, known as SAT schemes for finite difference approximations and penalties for spectral ones, and ours share similar properties but in our case one needs to pass at the interface a smaller amount of data than previously known schemes. This is important for multi-block parallelizations in several dimensions, for it implies that one obtains the same solution quality while sharing among different computational grids only a fraction of the data one would need for a comparable (in accuracy) SAT or Mattsson et al.'s scheme. The semi-discrete approximation used here preserves the norm and uses standard finite-difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge-Kutta method. This is crucial, since the explicit Runge-Kutta method would be impractical given the severe restrictions that arise from the stiff parts of the equations.