## Preprint 2017-30

## Tonatiuh Sanchez-Vizuet, Manuel Solano:

### A Hybridizable Discontinuous Galerkin solver for the Grad-Shafranov equation

### 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, (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration, and (5) is ideally suited for parallel implementations. 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, NSTX, ASDEX upgrade, and Field Reversed Configurations).

This preprint gave rise to the following definitive publication(s):

**Tonatiuh SANCHEZ-VIZUET, Manuel SOLANO: ***A hybridizable discontinuous Galerkin solver for the Grad-Shafranov equation*. Computer Physics Communications, vol. 235, pp.120-132, (2019).