Numerical Approach of Fast Free-boundary Equilibrium Solver Based on Finite Difference Method in Tokamaks

X. Song, E. Schuster, B. Leard, S.-T. Paruchuri, Z. Wang, L. Yang

11th ITER International School on “ITER Plasma Scenarios and Control”

San Diego, California, USA, July 25-29, 2022

Plasma equilibrium, a fundamental property in tokamaks, is widely used in exploring nominal plasma scenarios and developing various control strategies, from startup to termination of a discharge pulse. The plasma equilibrium equation in axisymmetric tokamaks, known as the Grad-Shafranov equation, can be solved either in an analytic way as a fixed-boundary solution or in a numerical way as a free-boundary solution. The free-boundary equilibrium (FBE) equation is a 2D (R, Z) partial differential equation (PDE) for the poloidal flux (ψ), which depends on the plasma current density, plasma pressure, and external coils currents. The dependence on the plasma current density, which in turn depends on ψ, makes the FBE equation nonlinear. In this work, a fast numerical approach based on the finite difference method associated with Dirichlet boundary conditions on a rectangular grid is developed to solve the FBE equation within a Picard iteration. During the iterations, plasma equilibrium quantities are varied to achieve prescribed plasma parameters, such as total Ip and βp. The solver can run in a direct way where external coils currents are given, and in an indirect way where reference plasma boundary is constrained to find the needed coils currents. To validate the mathematical correctness and accuracy of the numerical solver, the equilibrium solutions are benchmarked against other numerical codes such as FEEQS.M and EFIT based on EAST and DIII-D experimental data.

*Supported by the US DOE under DE-SC0021385.