Spectral Elements for Anisotropic Diffusion and Incompressible MHD

Paul Fischer, ANL

As the role of scientific simulation continues to expand with ever-increasing advances in computer hardware and numerical algorithms, high-order numerical discretizations are finding increased use in a diverse range of applications featuring multiphysics and multiscale phenomena. The interest in high-order methods stems from their attractive convergence properties and from the fact that there are now well-established ways to implement stable and efficient high-order schemes in complex domains. This talk provides a brief overview of one such discretization, the spectral element method, which combines the geometric flexibility of the finite element method with the rapid convergence and tensor-product efficiencies of global spectral methods. The primary focus will be application of the spectral element method to problems relating to fusion simulation, including anisotropic diffusion in a tokamak geometry and recent developments in spectral element simulation of liquid-metal MHD flows.

In tokamaks and related stellarator geometries, cross-field (radial) diffusion can be as much as $109$ weaker than diffusion parallel to the dominant magnetic field lines, which are helically wrapped on toroidal surfaces in the domain. The degree of anisotropy strongly influences the onset of the tearing mode instability, which leads to increased radial conductivity and thermal loss in the plasma. We show that high-order spatial discretizations can effectively treat anisotropic diffusion while low-order methods cannot. We explain this difference by considering a related eigenproblem.

We close with remarks regarding a recently initiated project in the simulation incompressible free-surface MHD flows.