Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains

TL;DR

This paper introduces a multidomain spectral-tau method for efficiently solving the three-dimensional helically reduced wave equation on complex two-center domains, with applications in astrophysics and general relativity.

Contribution

It presents the first 3D multidomain implementation of a spectral-tau method with sparse matrix realizations for the helically reduced wave equation.

Findings

Achieved sparse banded matrix representations of operators.

Demonstrated convergence of the global Schwarz-GMRES solver.

Potential applicability to other elliptic and mixed-type problems.

Abstract

We describe a multidomain spectral-tau method for solving the three-dimensional helically reduced wave equation on the type of two-center domain that arises when modeling compact binary objects in astrophysical applications. A global two-center domain may arise as the union of Cartesian blocks, cylindrical shells, and inner and outer spherical shells. For each such subdomain, our key objective is to realize certain (differential and multiplication) physical-space operators as matrices acting on the corresponding set of modal coefficients. We achieve sparse banded realizations through the integration "preconditioning" of Coutsias, Hagstrom, Hesthaven, and Torres. Since ours is the first three-dimensional multidomain implementation of the technique, we focus on the issue of convergence for the global solver, here the alternating Schwarz method accelerated by GMRES. Our methods may prove relevant for numerical solution of other mixed-type or elliptic problems, and in particular for the generation of initial data in general relativity.