A Hybridizable Discontinuous Galerkin solver for the Grad-Shafranov equation

TL;DR

This paper introduces a high-order Hybridizable Discontinuous Galerkin solver for the Grad-Shafranov equation, enabling accurate modeling of magnetic equilibria in fusion reactors with complex geometries and semi-linear characteristics.

Contribution

The paper presents a novel high-order HDG method for the Grad-Shafranov equation, including a new approach for piecewise smooth geometries and an efficient fixed-point iteration for semi-linearity.

Findings

High convergence order for flux and gradient

Effective handling of complex geometries with non-smooth boundaries

Validated with realistic fusion device configurations

Abstract

In axisymmetric fusion reactors, the equilibrium magnetic configuration can be expressed in terms of the solution to a semi-linear elliptic equation known as the Grad-Shafranov equation, the solution of which determines the poloidal component of the magnetic field. When the geometry of the confinement region is known, the problem becomes an interior Dirichlet boundary value problem. We propose a high order solver based on the Hybridizable Discontinuous Galerkin method. The resulting algorithm (1) provides high order of convergence for the flux function and its gradient, (2) incorporates a novel method for handling piecewise smooth geometries by extension from polygonal meshes, (3) can handle geometries with non-smooth boundaries and x-points, and (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration. The effectiveness of the algorithm is verified with computations for cases where analytic solutions are known on configurations similar to those of actual devices (ITER with single null and double null divertor, NSTX, ASDEX upgrade, and Field Reversed Configurations).