Efficient high-order singular quadrature schemes in magnetic fusion

TL;DR

This paper introduces a fast, high-order quadrature scheme for accurately computing singular integrals in magnetic fusion problems, improving efficiency and accuracy over traditional low-order methods.

Contribution

The authors develop a novel high-order quadrature method for singular integrals on toroidal surfaces, with publicly available code for community use.

Findings

Demonstrates high-order convergence in numerical examples

Achieves computational efficiency improvements

Provides a publicly released implementation

Abstract

Several problems in magnetically confined fusion, such as the computation of exterior vacuum fields or the decomposition of the total magnetic field into separate contributions from the plasma and the external sources, are best formulated in terms of integral equation expressions. Based on Biot-Savart-like formulae, these integrals contain singular integrands. The regularization method commonly used to address the computation of various singular surface integrals along general toroidal surfaces is low-order accurate, and therefore requires a dense computational mesh in order to obtain sufficient accuracy. In this work, we present a fast, high-order quadrature scheme for the efficient computation of these integrals. Several numerical examples are provided demonstrating the computational efficiency and the high-order accurate convergence. A corresponding code for use in the community has been publicly released.