A fast, high-order solver for the Grad-Shafranov equation

TL;DR

This paper introduces a rapid, high-order numerical solver for the Grad-Shafranov equation, enabling precise computation of plasma equilibria in toroidal geometries with complex profiles and large Shafranov shifts.

Contribution

The authors develop a novel solver combining conformal mapping, Fourier, and integral equation methods to achieve spectral accuracy for plasma equilibrium calculations.

Findings

Achieves high-order accuracy and spectral convergence.

Handles arbitrary plasma shapes and profiles.

Successfully computes equilibria with large Shafranov shifts.

Abstract

We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.