A fast, accurate, and easy to implement Kapur-Rokhlin quadrature scheme for singular integrals in axisymmetric geometries

TL;DR

This paper presents a Kapur-Rokhlin quadrature scheme that is fast, accurate, and easy to implement for calculating singular integrals in axisymmetric geometries, improving efficiency and precision in magnetic confinement fusion applications.

Contribution

The paper introduces a high-order, straightforward quadrature method specifically designed for logarithmic singularities in axisymmetric geometries, simplifying implementation and enhancing accuracy.

Findings

The scheme effectively handles logarithmic singularities in axisymmetric integrals.

It achieves high-order accuracy with ease of implementation.

Application to magnetic field calculations demonstrates improved efficiency.

Abstract

Many applications in magnetic confinement fusion require the efficient calculation of surface integrals with singular integrands. The singularity subtraction approaches typically used to handle such singularities are complicated to implement and low order accurate. In contrast, we demonstrate that the Kapur-Rokhlin quadrature scheme is well-suited for the logarithmically singular integrals encountered for a toroidally axisymmetric confinement system, is easy to implement, and is high order accurate. As an illustration, we show how to apply this quadrature scheme for the efficient and accurate calculation of the normal component of the magnetic field due to the plasma current on the plasma boundary, via the virtual casing principle.