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.

半边界的新颖映射 $（\mathrm{[R}，ž）$有限域到有限计算域用于解决托卡马克人的自由边界轴对称平衡问题。使用这种新的映射技术，仅使用“内部迭代”即可求解非线性Grad-Shafranov（GS）方程，而实际边界条件为无穷大。消除基于（例如）冯·哈格诺夫（Von Hagenow）方法的传统算法的外部迭代，可以使对自由边界问题的计算更加简单明了。通过将我们的数值解与线性GS方程的几种解析解进行比较，证明了我们方法的准确性。通常用于求解非线性GS方程的Picard迭代和Newton迭代都可能出现收敛问题。我们表明，通过在混合方案中一起使用两个方案，可以缓解每个问题。当使用适当的混合参数时，混合方案也显示为更快。这些方法使用开源在交易二平衡（DTEQ）代码中实现Deal-II有限元库。DTEQ的可靠性通过各种放电方案的实际实验数据得到证明。通过与基于Von Hagenow方法的解决方案进行比较，可以验证我们的映射技术的结果，并显示出更快的计算时间。