Spectral methods for the wave equation in second-order form

TL;DR

This paper introduces a new penalty spectral method for second-order wave equations that enhances stability and efficiency, potentially enabling direct spectral simulation of Einstein's equations in second-order form.

Contribution

A novel penalty method for second-order wave equations using Legendre polynomials, with proven stability and demonstrated numerical convergence.

Findings

Proven semi-discrete stability for the scalar wave equation in flat space.

Numerical demonstrations of stability and convergence in multi-domain settings.

Applicability to curved backgrounds and potential for Einstein's equations simulation.

Abstract

Current spectral simulations of Einstein's equations require writing the equations in first-order form, potentially introducing instabilities and inefficiencies. We present a new penalty method for pseudo-spectral evolutions of second order in space wave equations. The penalties are constructed as functions of Legendre polynomials and are added to the equations of motion everywhere, not only on the boundaries. Using energy methods, we prove semi-discrete stability of the new method for the scalar wave equation in flat space and show how it can be applied to the scalar wave on a curved background. Numerical results demonstrating stability and convergence for multi-domain second-order scalar wave evolutions are also presented. This work provides a foundation for treating Einstein's equations directly in second-order form by spectral methods.