Adaptive hp-Refinement for Spectral Elements in Numerical Relativity

TL;DR

This paper introduces a fully adaptive hp-refinement method for spectral element simulations in numerical relativity, improving accuracy and efficiency in handling problems with multiple scales.

Contribution

It presents a novel, physics-agnostic hp-refinement algorithm integrated into the bamps code, with detailed implementation and validation in various test problems.

Findings

Enhanced accuracy over static meshes

Improved computational efficiency and scaling

Successful application to complex wave equations

Abstract

When a numerical simulation has to handle a physics problem with a wide range of time-dependent length scales, dynamically adaptive discretizations can be the method of choice. We present a major upgrade to the numerical relativity code bamps in the form of fully adaptive, physics-agnostic hp-refinement. We describe the foundations of mesh refinement in the context of spectral element methods, the precise algorithm used to perform refinement in bamps, as well as several indicator functions used to drive it. Finally, we test the performance, scaling, and the accuracy of the code in treating several 1d and 2d example problems, showing clear improvements over static mesh configurations. In particular, we consider a simple non-linear wave equation, the evolution of a real scalar field minimally coupled to gravity, as well as nonlinear gravitational waves.