A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics
<|reference_start|>A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics: 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.<|reference_end|>
@article{dao2024a,
    title={A nodal based high order nonlinear stabilization for finite element
  approximation of Magnetohydrodynamics},
    author={Tuan Anh Dao and Murtazo Nazarov},
    journal={arXiv preprint arXiv:2404.09311},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.09311},
    primaryClass={math.NA cs.NA}
}