General relativistic null-cone evolutions with a high-order scheme

TL;DR

This paper introduces a high-order numerical scheme for solving Einstein's equations on null hypersurfaces, improving accuracy and efficiency in modeling gravitational radiation in asymptotically flat spacetimes.

Contribution

It develops a 4th-order radial integration and spectral angular representation scheme for characteristic evolution of Einstein's equations, demonstrating stability and convergence.

Findings

Scheme achieves higher accuracy with lower computational cost.

Numerical stability and convergence are confirmed.

Suitable for detailed gravitational wave studies.

Abstract

We present a high-order scheme for solving the full non-linear Einstein equations on characteristic null hypersurfaces using the framework established by Bondi and Sachs. This formalism allows asymptotically flat spaces to be represented on a finite, compactified grid, and is thus ideal for far-field studies of gravitational radiation. We have designed an algorithm based on 4th-order radial integration and finite differencing, and a spectral representation of angular components. The scheme can offer significantly more accuracy with relatively low computational cost compared to previous methods as a result of the higher-order discretization. Based on a newly implemented code, we show that the new numerical scheme remains stable and is convergent at the expected order of accuracy.