一种用于可压缩电阻性霍尔磁流体动力学系统的保结构数值方法
A structure-preserving Numerical Method for the Compressible Resistive-Hall-MHD System
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中文总结 AI 辅助
研究可压缩电阻性霍尔磁流体动力学系统，提出保结构数值方法，将微分算子拆分，对不同部分采用特定有限元及求解方法，经强制性估计等改进，通过求解多个基准问题验证方法的鲁棒性和准确性。
中文摘要 AI 辅助
本文提出了一种用于可压缩电阻性霍尔磁流体动力学（MHD）模型的保结构方法。微分算子被拆分为两部分：由可压缩欧拉方程组成的流体动力学部分，以及由耦合洛伦兹力和感应方程的系统组成的磁学部分。该方法对欧拉部分使用连续拉格朗日元，对磁学部分使用旋度协调有限元空间。流体动力学部分保持密度和内能的正性、总能量守恒以及比熵的最小原理。由于有限元的选择，磁学部分保持散度对合约束。流体部分用显式强稳定性保持龙格 - 库塔（SSP - RK）方法求解，磁学部分用克兰克 - 尼科尔森方法求解，这需要使用牛顿法。给出了相应牛顿迭代雅可比矩阵的强制性估计。引入高阶人工电阻率以改善非线性残差的条件和雅可比矩阵的可逆性。通过求解几个具有挑战性的基准问题，包括平滑哨声波、用于比较电阻性MHD和电阻性霍尔MHD的奥尔扎克 - 唐涡旋以及磁重联问题，来验证该方法的鲁棒性和准确性。
英文摘要
In this paper, we present a structure-preserving method for the compressible resistive Hall-magnetohydrodynamics (MHD) model. The differential operator is split into two parts: a hydrodynamic part consisting of the compressible Euler equations, and a magnetic part consisting of a system coupling the Lorentz force and the induction equation. The method uses continuous Lagrange elements for the Euler part and a curl-conforming finite element space for the magnetic part. The hydrodynamic part preserves the positivity of the density and internal energy, the conservation of total energy, and the minimum principle for the specific entropy. Owing to the choice of finite elements, the magnetic part preserves the divergence involution constraint. The fluid part is solved using explicit strong-stability-preserving Runge-Kutta (SSP-RK) methods, whereas the magnetic part is solved by Crank-Nicholson method, which requires using Newton's method. Coercivity estimates for the Jacobian of the corresponding Newton iteration are presented. We introduce a high-order artificial resistivity to improve the conditioning of the nonlinear residual and the invertibility of the Jacobian. Several challenging benchmarks, including a smooth whistler wave, the Orszag-Tang vortex for comparing resistive MHD with resistive Hall-MHD, and a magnetic reconnection problem, are solved to validate the robustness and accuracy of the method.
