Toward a Consistent Framework for High Order Mesh Refinement Schemes in Numerical Relativity

TL;DR

This paper introduces a fourth order stable mesh refinement scheme for numerical relativity that avoids buffer zones, reduces spurious reflections, and ensures consistency with high order Runge Kutta methods.

Contribution

It presents a novel high order mesh refinement algorithm with explicit boundary data and transition zones, improving stability and accuracy in numerical relativity simulations.

Findings

Significant reduction in spurious reflections at refinement boundaries

Stable fourth order scheme compatible with Runge Kutta methods

Effective handling of wave propagation across mesh boundaries

Abstract

It has now become customary in the field of numerical relativity to couple high order finite difference schemes to mesh refinement algorithms. To this end, different modifications to the standard Berger-Oliger adaptive mesh refinement algorithm have been proposed. In this work we present a fourth order stable mesh refinement scheme with sub-cycling in time for numerical relativity. We do not use buffer zones to deal with refinement boundaries but explicitly specify boundary data for refined grids. We argue that the incompatibility of the standard mesh refinement algorithm with higher order Runge Kutta methods is a manifestation of order reduction phenomena, caused by inconsistent application of boundary data in the refined grids. Our scheme also addresses the problem of spurious reflections that are generated when propagating waves cross mesh refinement boundaries. We introduce a transition zone on refined levels within which the phase velocity of propagating modes is allowed to decelerate in order to smoothly match the phase velocity of coarser grids. We apply the method to test problems involving propagating waves and show a significant reduction in spurious reflections.