Multidomain Galerkin-Collocation method: characteristic spherical collapse of scalar fields

TL;DR

This paper develops a spectral domain decomposition approach using the Galerkin-Collocation method to simulate spherical scalar field collapse, successfully reproducing critical phenomena like Choptuik's scaling law.

Contribution

It introduces a systematic multidomain Galerkin-Collocation framework for hyperbolic problems, specifically applied to scalar field collapse in spherical symmetry.

Findings

Successfully reproduces Choptuik's scaling law

Captures oscillatory component due to discrete self-similarity

Validates the multidomain spectral method for hyperbolic problems

Abstract

We initiate a systematic implementation of the spectral domain decomposition technique with the Galerkin-Collocation (GC) method in situations of interest such as the spherical collapse of a scalar field in the characteristic formulation. We discuss the transmission conditions at the interface of contiguous subdomains that are crucial for the domain decomposition technique for hyperbolic problems. We implemented codes with an arbitrary number of subdomains, and after validating them, we applied to the problem of critical collapse. With a modest resolution, we obtain the Choptuik's scaling law and its oscillatory component due to the discrete self-similarity of the critical solution.