High-Resolution Finite Element Schemes for the Compressible MHD Equations

本项目的目标是开发和分析在非结构网格上求解理想磁流体力学(MHD)方程的高分辨率有限元格式。主要的焦点是在离散水平上设计满足所有特定问题的物理约束(守恒定律、最大值原理、无发散条件)的算法。该方法属于交错约束运输(CT)方法家族。物理上兼容的节点、边和面有限元空间的选择使避免导致数值不稳定的发散误差成为可能。利用一种新型的人工耗散和基于单元的通量校正输运(FCT)算法，构造了可压缩MHD系统的无振荡近似。守恒量的连续节点近似被用来从欧姆定律和麦克斯韦-法拉第定律计算基于边的电场强度和基于面的磁通密度。使用定制的限制策略可防止虚假欠冲/过冲的形成。所提出的MHD求解器可以在隐式或显式模式下运行。它不需要维分裂，并且产生螺线管磁场，而不需要解向量值磁势的泊松方程或输运方程。开发的软件将用于研究理想化的磁Z箍缩衬里内爆和三维磁瑞利-泰勒(MRT)不稳定性。

The goal of this project is the development and analysis of high-resolution finite element schemes for solving the equations of ideal magnetohydrodynamics (MHD) on unstructured meshes in 3D. The main focus is on the design of algorithms satisfying all problem-specific physical constraints (conservation laws, maximum principles, divergence-free conditions) at the discrete level. The proposed approach belongs to the family of staggered constrained transport (CT) methods. A physics-compatible choice of nodal, edge, and face finite element spaces makes it possible to avoid divergence errors that cause numerical instabilities. A nonoscillatory approximation to the compressible MHD system is constructed using a new type of artificial dissipation and an element-based version of the flux-corrected transport (FCT) algorithm. The continuous nodal approximation of the conserved quantities is used to calculate the edge-based electric field intensity and the face-based magnetic flux density from the Ohm and Maxwell-Faraday laws. The formation of spurious undershoots/overshoots is prevented using a custom-made limiting strategy. The proposed MHD solver can be run in an implicit or explicit mode. It does not require dimensional splitting and produces solenoidal magnetic fields without the cost of solving a Poisson equation or a transport equation for the vector-valued magnetic potential. The developed software will be used to study idealized magnetic Z-pinch liner implosions and 3D magnetic Rayleigh-Taylor (MRT) instabilities.