A free-boundary equilibrium solver with a hybrid iteration method in a semi-bounded computational domain

Abstract

A novel mapping of the semi-bounded (R,Z) domain to a finite computational domain is used to solve the free-boundary axisymmetric equilibrium problem for tokamaks. Using this new mapping technique, the nonlinear Grad–Shafranov (GS) equation can be solved using only the “inner iterations” but with the actual boundary condition at infinity. Eliminating the outer iterations of the traditional algorithms based on, for example, Von Hagenow’s method can make the calculation for the free-boundary problem much more straightforward and easier. The accuracy of our methods is demonstrated by comparing our numerical solutions with several analytic solutions of linear GS equations. Both Picard and Newton iterations commonly used to solve the nonlinear GS equation can exhibit convergence problems. We show that each problem can be alleviated by using the two schemes together in a hybrid scheme. The hybrid scheme is also shown to be faster when an appropriate blending parameter is used. These methods are implemented in the Deal Two Equilibrium (DTEQ) code using the open-source deal-II finite element library. The reliability of DTEQ is demonstrated using actual experimental data for various discharge regimes. The results of our mapping technique are validated by comparing with solutions based on Von Hagenow’s method, as well as showing much faster computation time.