Accurate Derivative Evaluation for any Grad-Shafranov Solver

TL;DR

This paper introduces a numerical scheme that accurately computes derivatives of solutions to Poisson and Grad-Shafranov equations, enhancing magnetic confinement fusion simulations by enabling precise magnetic field and current density evaluations on coarse grids.

Contribution

A novel numerical method that combines with existing solvers to compute derivatives with the same convergence order as the solution, improving accuracy in fusion simulation quantities.

Findings

Enables high-accuracy derivative computation on coarse grids

Improves evaluation of magnetic field and current density

Compatible with any fixed boundary finite element 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.