Summation by Parts and Truncation Error Matching on Hyperboloidal Slices

TL;DR

This paper investigates the stability and effectiveness of summation by parts schemes on hyperboloidal slices for wave equations, demonstrating long-term stability and energy conservation, with potential applications in gravitational waveform extraction.

Contribution

It introduces a stable SBP scheme with truncation error matching on hyperboloidal slices, suitable for gravitational waveform simulations, and analyzes its performance for different field types.

Findings

Long-term stability and norm convergence for massless scalar fields

Excellent energy conservation observed even with massive fields

Scheme can be generalized to higher order or spectral accuracy

Abstract

We examine stability of summation by parts (SBP) numerical schemes that use hyperboloidal slices to include future null infinity in the computational domain. This inclusion serves to mitigate outer boundary effects and, in the future, will help reduce systematic errors in gravitational waveform extraction. We also study a setup with truncation error matching. Our SBP-Stable scheme guarantees energy-balance for a class of linear wave equations at the semidiscrete level. We develop also specialized dissipation operators. The whole construction is made at second order accuracy in spherical symmetry, but could be straightforwardly generalized to higher order or spectral accuracy without symmetry. In a practical implementation we evolve first a scalar field obeying the linear wave equation and observe, as expected, long term stability and norm convergence. We obtain similar results with a potential term. To examine the limitations of the approach we consider a massive field, whose equations of motion do not regularize, and whose dynamics near null infinity, which involve excited incoming pulses that can not be resolved by the code, is very different to that in the massless setting. We still observe excellent energy conservation, but convergence is not satisfactory. Overall our results suggest that compactified hyperboloidal slices are likely to be provably effective whenever the asymptotic solution space is close to that of the wave equation.