Accurate derivative evaluation for any Grad–Shafranov solver

Abstract

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.

An adaptive fast multipole accelerated Poisson solver for complex geometries

“…See, for instance, P. Farkas' PhD thesis [23] and the approach of [41] which leverages the Sherman-Lauricella integral equations for elasticity. There are two main reasons to not consider extrapolations based on polyharmonic equations of higher order than the biharmonic equation: (1) very few numerical tools have been developed for such equations and (2) the equations require to provide high order derivatives of the data f in the direction normal to the boundary, which in most physical applications are not readily available, and can be challenging to compute with high accuracy numerically, even when using integral equation based methods [24]. For our numerical tests, and in the version of the code which will be available online, only the harmonic expansion calculated by solving (19) is implemented.…”

“…We resolve the difficulty by computing the integrals for points near the boundary using the quadrature-by-expansion (QBX) method. We will not present the fundamentals of the QBX scheme here, since clear presentations for situations very closely related to the one we encounter here can be found in [21,24,42]. We will however stress two modifications to the standard QBX scheme which we implemented in our solver.…”

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

“…However, there typically is a price to pay for these advantages. Integral formulations often involve singular integrands, which are subtle to handle numerically (Freidberg et al 1976;Merkel 1986;Atkinson 1997;Chance 1997;Ludwig et al 2006Ludwig et al , 2013Klöckner et al 2013;Kress 2014;Landreman & Boozer 2016;Ricketson et al 2016;Malhotra et al 2019a). The numerical difficulty of integrating these singular integrands depends on the nature of the singularity, the distribution of sources and the relative location of the evaluation points (often known as target points or observation points) with respect to the sources.…”

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

“…However, there typically is a price to pay for these advantages. Integral formulations often involve singular integrands, which are subtle to handle numerically (Freidberg et al 1976;Merkel 1986;Atkinson 1997;Chance 1997;Ludwig et al 2006Ludwig et al , 2013Klöckner et al 2013;Kress 2014;Ricketson et al 2016;Malhotra et al 2019a;Landreman & Boozer 2016). The numerical difficulty of integrating these singular integrands depends on the nature of the singularity, the distribution of sources, and the relative location of the evaluation points (often known as target points or observation points) with respect to the sources.…”