Properties of four numerical schemes applied to a scalar nonlinear scalar wave equation with a GR-type nonlinearity

TL;DR

This paper compares four numerical schemes for scalar nonlinear wave equations, analyzing their stability, dispersion, and dissipation properties, and identifies the 4th order Runge-Kutta as the most balanced method.

Contribution

It provides a detailed stability and dispersion analysis of four numerical schemes applied to nonlinear wave equations relevant to general relativity.

Findings

MOL schemes are most dispersive and dissipative.

CFLN scheme is most accurate but less dissipative.

4th order Runge-Kutta offers the best compromise.

Abstract

We study stability, dispersion and dissipation properties of four numerical schemes (Iterative Crank-Nicolson, 3'rd and 4'th order Runge-Kutta and Courant-Fredrichs-Levy Non-linear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar non-linear wave equation with a type of non-linearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant-Fredrichs-Levy Non-linear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4'th order Runge-Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.