Exploring Critical Collapse in the Semilinear Wave Equation using Space-Time Finite Elements

TL;DR

This paper introduces an adaptive space-time finite element method to study critical collapse in the semilinear wave equation, enabling exploration of self-similar solutions with improved numerical techniques.

Contribution

It presents a novel adaptive space-time finite element approach with parallelization for analyzing critical collapse in the semilinear wave equation, advancing beyond previous finite difference methods.

Findings

Effective mesh refinement improves solution accuracy.

The time additive Schwarz preconditioner enhances computational efficiency.

Self-similar solutions are successfully identified in 1+1 and 2+1 dimensions.

Abstract

A fully implicit numerical approach based on the space-time finite element method is implemented for the semilinear wave equation in 1(space) + 1(time) and 2 + 1 dimensions to explore critical collapse and search for self-similar solutions. Previous work studied this behavior by exploring the threshold of singularity formation using time marching finite difference techniques while this work introduces an adaptive time parallel numerical method to the problem. The semilinear wave equation with a $p = 7$ term is examined in spherical symmetry. The impact of mesh refinement and the time additive Schwarz preconditioner in conjunction with Krylov Subspace Methods are examined.