A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

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 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 spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.

Automated summary

This paper presents a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method utilizes continuous Lagrange polynomials and explicit Runge-Kutta schemes to achieve accurate results. By tracking the shock and discontinuity positions, and adding sufficient viscosity, sharp shocks and discontinuities are resolved effectively.