A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics

TL;DR

This work develops a nodal-based, high-order stabilization for finite element magnetohydrodynamics by combining a parameter-free, mesh-dependent first-order viscosity with a residual-based high-order diffusion that activates near shocks. A two-mesh construction is used to build the viscosity coefficients on nodal values while conserving the high-order space, and a simple divergence-cleaning step maintains practical stability. The method employs explicit Runge-Kutta time stepping and demonstrates robust shock-capturing and high-order accuracy on standard MHD benchmarks (e.g., Brio–Wu, Orszag–Tang, Kelvin–Helmholtz, and blast problems) while proving a discrete maximum principle for scalar laws. Although positivity is not guaranteed, the framework provides a scalable, high-order, stabilization mechanism that can be extended with limiting techniques for positivity-preserving MHD simulations in the future.

Abstract

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.

Figures (16)

• Figure 1: Nodal distribution and sub-meshes for different polynomial spaces. The green dashed area is the support of ${\boldsymbol N}_i$ in the corresponding ${\mathbb P}_1$ sub-mesh.
• Figure 2: First-order solutions to the Brio-Wu problem and the corresponding amounts of artificial viscosity. The reference solution is produced by the Athena code Stone_2008 using 10001 grid points.
• Figure 3: Convergence of the numerical solutions to the Brio-Wu problem. The high-order solution. The reference solution is produced by the Athena code Stone_2008 using 10001 grid points.
• Figure 4: The amount of artificial viscosity added to the numerical solutions in the Brio-Wu problem. The high-order viscosity $\varepsilon^{\text{RV}}_h$ is plotted together with its first-order upper bound $\varepsilon^L_h$.
• Figure 5: ${\mathbb P}_3$ solutions to Orszag-Tang problem at time $t=0.5$ on $200\times 200$${\mathbb P}_1$ and ${\mathbb P}_3$ nodes.

Theorems & Definitions (8)

• Definition 3.1: Nodal artificial viscosity
• Remark 3.1: Relation to upwind schemes
• Remark 3.2: Scaling of the residual
• Lemma A.1: Guermond_Nazarov_2014
• Theorem A.1
• proof
• Remark A.1
• Remark A.2