Efficient high-order singular quadrature schemes in magnetic fusion

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 1 . *

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

“…We shall provide an explicit expression in section 4.1, as we treat in detail the application of our method to the calculation of the virtual casing principle. At this point, we just highlight the fact that the singularity in the Green's function when x = y requires the use of specialized quadrature when the target x is located on the surface Γ , or regularization methods (Freidberg et al 1976;Merkel 1986;Chance 1997;Landreman & Boozer 2016;Drevlak et al 2018;Malhotra et al 2019b). The purpose of this article is to show that for applications in axisymmetric geometries, the Kapur-Rokhlin quadrature scheme is simpler to implement than the known regularization methods used in the magnetic confinement community, and leads to high order convergence.…”

“…However, it also has the disadvantage that one must use a careful principal value integration procedure to interpret the virtual casing principle for x ∈ Γ , as discussed in theorem 2 and its proof in the appendix. The high-order singularity cancellation quadrature scheme we recently proposed for singular integrals on general non-axisymmetric surfaces (Malhotra et al 2019b) automatically yields the appropriate principal value of the integral. However, it does so thanks to the intrinsic two-dimensional nature of the integral.…”

“…The plasma equilibrium we consider is the Solov'ev equilibrium described above (Lütjens et al 1996;. Since we do not know the analytic solution to this problem, we compare our implementation with an existing high order accurate implementation from Malhotra et al (2019b). The method used in that implementation is different from the approach presented here in several ways, making it appropriate for our verification.…”

“…In this article, we will focus on the common situation in which we are trying to evaluate layer potentials at the source locations. This is for example the standard situation when applying Green's identity (Freidberg et al 1976;Merkel 1986;Pustovitov 2008;Malhotra et al 2019b).…”

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

“…For non-axisymmetric applications, we recently presented an efficient high-order quadrature scheme based on a different approach (Malhotra et al. 2019 b ), and alternative schemes may also provide good performance (Bruno & Garza 2020; Wu & Martinsson 2021). However, for axisymmetric cases, none of these schemes reduce to as simple and efficient a method as the Kapur–Rokhlin approach we present here.…”