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.

翻译：我们提出了一种新型高阶节点人工黏性方法，用于求解磁流体动力学（MHD）方程。与传统方法不同，该方法无需引入特定参数。黏性项依赖于网格，但无需显式定义网格尺寸。该方法采用多重网格策略：黏性系数基于细网格上构造的线性多项式空间建立，该空间与有限元逼近空间的节点值相对应。利用MHD方程的残差，在激波和间断附近以局部化方式引入高阶黏性，从而精确捕捉并解析激波。随后采用高阶龙格-库塔方法进行时间离散。通过一系列具有挑战性的基准测试问题，验证了所提方法在求解MHD方程时的鲁棒性与高阶精度。