Stabilized bi-cubic Hermite-Bézier finite element method: application to gas-plasma interactions in tokamaks

Abstract:
Development of a computational solver based upon the high-order, high-resolution Galerkin finite element method (FEM) often encounters two challenges: First, the Galerkin FEMs give central approximations to the differential operators and their use in the simulation of the convection-dominated flows may lead to the dispersion errors yielding entirely wrong numerical solutions. Secondly, high-order, high-resolution numerical methods are known to produce high wave-number oscillations in the vicinity of shocks/discontinuities in the numerical solution adversely affecting the stability of the method. In this work, we present the stabilized finite element method for plasma fluid models to address the two challenges. The numerical stabilization is based on two strategies: Variational Multiscale (VMS) and the shock-capturing approach. The former strategy takes into account (the approximation of) the effect of the unresolved scales onto resolved scales to introduce upwinding in the Galerkin FEM. The latter adaptively adds artificial viscosity only in the vicinity of shocks. These two strategies can be used to improve the stability and robustness of the numerical methods that are used to solve a wide range of physical problems in fluid dynamics, plasma physics, astrophysics, etc.

In this work, we demonstrate the use of stabilized FEM to solve challenging applications in the tokamak plasma. The physical phenomena involved are convection dominated, anisotropic, and highly nonlinear, contain shocks, and may also contain strong local sources. First, I will present the bi-cubic Hermite Bézier FEM that is implemented in the nonlinear MHD code JOREK and associated numerical challenges. Then I will present the full magnetohydrodynamics (MHD) model that we implemented in JOREK to extend its capabilities from the reduced MHD modeling. Finally, I will demonstrate the use of the developed stabilized FEM to perform simulations of complex gas-plasma interactions (GPI) occurring in Massive material injection (MMI) experiments in tokamaks.