Finite Difference Methods for Second Order in Space, First Order in Time Hyperbolic Systems and the Linear Shifted Wave Equation as a Model Problem in Numerical Relativity

TL;DR

This paper investigates high-order finite difference methods for hyperbolic systems, focusing on stability, accuracy, and the effects of centered versus off-centered schemes, with applications to the shifted wave equation in numerical relativity.

Contribution

It extends stability analysis to high-order schemes and compares centered and off-centered discretizations for hyperbolic PDEs, especially the shifted wave equation.

Findings

Off-centered schemes can improve accuracy over centered schemes.

Stability properties are extended to arbitrary order accuracy.

Analysis of numerical phase and group speeds for the shifted wave equation.

Abstract

Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order derivatives that can be identified with advection terms are approximated with non-centered finite difference operators.We first derive general properties of these discrete operators, then we extend a known result on numerical stability for such systems to general order of accuracy.As an application we analyze the shifted wave equation, including the behavior of the numerical phase and group speeds at different orders of approximations. Special attention is paid to when the use of off-centered schemes improves the accuracy over the centered schemes.