We present a novel technique for imposing non-linear entropy conservative and entropy stable wall boundary conditions for the resistive magnetohydrodynamic equations in the presence of an adiabatic wall or a wall with a prescribed heat entropy flow, addressing three scenarios: electrically insulating walls, thin walls with finite conductivity, and perfectly conducting walls. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts, and simultaneous-approximation-term operators. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions is coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators) on high-order unstructured grids are used to demonstrate the new procedure's accuracy, robustness, and efficacy for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, nodal and modal discontinuous Galerkin, and flux reconstruction schemes.

翻译：本文提出了一种新颖的技术，用于在绝热壁面或具有规定热熵流的壁面条件下，为电阻磁流体动力学方程施加非线性熵守恒与熵稳定的壁面边界条件，涵盖了三种情形：电绝缘壁面、具有有限电导率的薄壁面以及理想导电壁面。该方法基于对角范数、分部求和与同步逼近项算子的形式化体系及拟态特性。通过采用线法，当所提出的边界条件数值施加方式与熵守恒或熵稳定的离散内部算子耦合时，可获得整个区域的半离散熵估计。所得估计模拟了在连续层面上获得的全局熵估计。壁面处的边界数据通过惩罚通量方法及同步逼近项技术，对守恒变量和熵变量的梯度进行弱施加。在高阶非结构网格上使用间断谱配点算子（集中质量节点间断伽辽金算子）验证了新方法在弱施加边界条件时的精度、鲁棒性和有效性。数值模拟证实了所提技术的非线性稳定性，并将其应用于三维流动。所述方法与任何对角范数分部求和空间算子兼容，包括有限元、有限差分、有限体积、节点与模态间断伽辽金以及通量重构格式。