A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

TL;DR

This paper develops a pseudospectral method using double Fourier expansion and spin-weighted spherical harmonic filtering for evolving tensor fields on a spherical shell, demonstrated on black hole simulations with high computational efficiency.

Contribution

It introduces a novel pseudospectral approach with spin-weighted filters for tensor PDEs on spheres, improving stability and efficiency in numerical relativity.

Findings

Successful implementation of spin-weighted spherical harmonic filter for stability.

Achieved up to 20x speed-up with GPU parallelization.

Validated method on black hole evolution simulations.

Abstract

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.