A comparison of spectral element and finite difference methods using statically refined nonconforming grids for the MHD island coalescence instability problem

TL;DR

This paper compares spectral element and finite difference methods on refined grids for simulating the MHD island coalescence instability, demonstrating spectral element's superior accuracy in resolving sharp current layers.

Contribution

It introduces a comparison framework for spectral element and finite difference methods using static adaptive grids in MHD simulations, highlighting spectral element's higher accuracy.

Findings

Spectral element methods achieve higher accuracy than finite difference methods.

Refined grid scaling is roughly linear with effective resolution.

Spectral element simulations can approach full spectral accuracy.

Abstract

A recently developed spectral-element adaptive refinement incompressible magnetohydrodynamic (MHD) code [Rosenberg, Fournier, Fischer, Pouquet, J. Comp. Phys. 215, 59-80 (2006)] is applied to simulate the problem of MHD island coalescence instability (MICI) in two dimensions. MICI is a fundamental MHD process that can produce sharp current layers and subsequent reconnection and heating in a high-Lundquist number plasma such as the solar corona [Ng and Bhattacharjee, Phys. Plasmas, 5, 4028 (1998)]. Due to the formation of thin current layers, it is highly desirable to use adaptively or statically refined grids to resolve them, and to maintain accuracy at the same time. The output of the spectral-element static adaptive refinement simulations are compared with simulations using a finite difference method on the same refinement grids, and both methods are compared to pseudo-spectral simulations with uniform grids as baselines. It is shown that with the statically refined grids roughly scaling linearly with effective resolution, spectral element runs can maintain accuracy significantly higher than that of the finite difference runs, in some cases achieving close to full spectral accuracy.