Collaborative Research: Advanced Numberical Techniques for the Simulation of Magnetohydrodynamics

本研究的总体目标是为复杂流体系统的模拟开发数值模型和有效的能量一致方法，特别是为磁流体动力学（MHD）系统获得物理精确的解决方案。这项工作的重点是最小化这些系统的近似解决方案的计算成本，以便为这类问题开发实际的模拟。底线是进一步发展离散化和求解算法，以产生每计算成本的最高准确性。这是通过推导MHD系统的离散能量定律并证明它们与连续数学模型一致来实现的。研究人员通过使用能量变分方法（EVA）来进一步研究MHD模型本身，并确定哪种“风味”的MHD方程是捕获相关物理的最简单模型。使用有限元方法求解复杂流体和电磁问题的主要问题之一是精确地保持无散度解。在这里，研究人员分析了精确满足这些数量的离散化方法，同时仍然适用于有效的解决方案。最后，目前使用的离散化方法的主要瓶颈是线性解的缓慢收敛。通过进一步发展多网格方法的偏微分方程系统，研究人员创建了一个鲁棒和有效的算法，这些复杂的系统。高效的MHD系统仿真框架的开发对聚变能的研究具有重大影响，因为该模型用于描述聚变反应堆中发生的各种现象，包括撕裂模式和锯齿不稳定性。随着法国的国际热核实验反应堆（ITER）和劳伦斯利弗莫尔国家实验室的国家点火装置（NIF）等项目试图获得可持续的核聚变能源，与这些项目相关的领域的科学计算至关重要。此外，正在开发的数值工具，如节能有限元离散化和pde系统的最优多网格求解器，适用于多物理场和多尺度系统中的各种其他问题。最后，该项目支持一名研究生，训练并使他们接触到最新的科学发现和与建模、离散化和解决等离子体物理和复杂流体计算建模问题有关的工具。

The overall objective of this research is to develop numerical models and efficient energy-consistent methods for simulation of complex fluid systems and to obtain physically-accurate solutions for magnetohydrodynamic (MHD) systems, in particular. This work focuses on minimizing the computational cost of approximating solutions to these systems in order to develop practical simulations for this type of problem. The bottom line is to further develop discretization and solution algorithms that yield the most accuracy per computational cost. This is achieved by deriving discrete energy laws for MHD systems and proving that they are consistent with the continuous mathematical model. The investigators accomplish this by using the Energetic-Variational Approach (EVA) to further investigate the MHD model itself and determining which "flavor" of the MHD equations is the simplest model that captures the relevant physics. One of the main issues with using finite-element methods for solving complex fluid and electromagnetic problems has been the precise preservation of divergence-free solutions. Here, the investigators analyze discretization methods that satisfy these quantities accurately, while remaining amenable to efficient solution. Finally, the main bottleneck in the discretization methods used so far is the slow convergence of the linear solvers. By further developing multigrid methods for systems of partial differential equations, the investigators create a robust and efficient algorithm for these complex systems.The development of an efficient simulation framework for MHD systems has a major impact on the study of fusion energy, as this model is used to describe various phenomena that occur in fusion reactors, including tearing mode and sawtooth instabilities. With projects such as the International Thermonuclear Experimental Reactor (ITER) in France and the National Ignition Facility (NIF) at Lawrence Livermore National Lab attempting to obtain sustainable fusion energy, scientific computing in fields related to these projects is critical. In addition, the numerical tools being developed, such as energy-preserving finite-element discretizations and optimal multigrid solvers for systems of PDEs, are applicable to a wide variety of other problems in multi-physics and multi-scale systems. Finally, the project supports a graduate student, training and exposing them to the latest scientific findings and tools related to modeling, discretization, and solution of problems in the computational modeling of plasma physics and complex fluids.