# Taylor States in Stellarators: A Fast High-order Boundary Integral Solver

## TL;DR

This paper introduces a fast, high-order boundary integral solver for calculating Taylor relaxed states in complex stellarator geometries, improving accuracy and efficiency in magnetic equilibrium computations.

## Contribution

It develops a well-conditioned second-kind boundary integral equation using the generalized Debye source formulation for stellarator magnetic field calculations.

## Key findings

- High accuracy demonstrated through numerical examples
- Efficient computation with spectral discretization and high-order quadrature
- Comparison shows competitive performance with leading existing codes

## Abstract

We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.

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Source: https://tomesphere.com/paper/1902.01205