A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics - Dao, Tuan Anh; Nazarov, Murtazo (Journal of computational physics, 2024)
We present a novel high-order nodal artificial viscosity approach designed for solving Magnetohydrodynamics (MHD) equations. Unlike conventional methods, our approach eliminates the need for ad hoc parameters. The viscosity is mesh-dependent, yet explicit definition of the mesh size is unnecessary. Our method employs a multimesh strategy: the viscosity coefficient is constructed from a linear polynomial space constructed on the fine mesh, corresponding to the nodal values of the finite element approximation space. The residual of MHD is utilized to introduce high-order viscosity in a localized fashion near shocks and discontinuities. This approach is designed to precisely capture and resolve shocks. Then, high-order Runge-Kutta methods are employed to discretize the temporal domain. Through a comprehensive set of challenging test problems, we validate the robustness and high-order accuracy of our proposed approach for solving MHD equations. •New nodal-based artificial viscosity method for MHD.•The method does not include any ad hoc parameters or explicit definition of the mesh size.•The viscosity coefficient is built in a multigrid strategy.•The method is proven to preserve positivity for scalar conservation laws using linear finite elements.