CRATOS-GS: A free-boundary, hierarchical adaptive mesh refinement Grad–Shafranov solver

We present CRATOS-GS, a free-boundary hierarchical adaptive mesh refinement (AMR) Grad–Shafranov solver, discretized on Cartesian meshes. The solver is suitable for stand-alone equilibrium calculations, as a foundational component of reduced modeling, and also as an integrated component of non-linear extended magnetohydrodynamic codes. The solver employs a geometric multigrid method to efficiently solve the underlying elliptic system for static inverse as well as forward problems. It supports a wide range of physics-based and geometric constraints, including optional plasma shape features (X-points, iso-flux surfaces, and fixed flux points), equilibrium properties (total plasma current and a choice of poloidal beta or central pressure), and multiple choices for defining coil connections and current bounds. The plasma region is delineated by a fast-marching algorithm, and a level-set function is used to define the arbitrarily complex geometry of the first wall, which allows for efficient calculation of limited plasma profiles. In addition, toroidal rotation is supported for spatially varying velocity fields, which enables the effects of real operational conditions, such as neutral beam injection, to be taken into account in the equilibrium state. In order to ensure reliable convergence, particularly for the forward problem, a vertical stabilization technique is implemented. CRATOS-GS is validated by means of a comprehensive suite of benchmarks, including fixed-boundary problems with analytic solutions and free-boundary case studies for realistic reactor configurations. Additional benchmarks are used to assess the efficacy of AMR in capturing steep gradients while reducing computational costs compared to unigrid calculations.