A high-order residual-based viscosity finite element method for the ideal MHD equations

TL;DR

This paper introduces a high-order, residual-based viscosity finite element method for ideal MHD equations that effectively captures shocks and discontinuities while maintaining stability and high accuracy.

Contribution

It develops a novel high-order shock-capturing finite element method using residual-based viscosity for ideal MHD, achieving high accuracy and stability.

Findings

Achieves up to fourth-order convergence on smooth problems.

Successfully captures shocks and discontinuities in various benchmark tests.

Demonstrates robustness and stability in multi-dimensional simulations.

Abstract

We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds a sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spatial dimensions. Sharp shocks and discontinuity resolutions are obtained.