A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Tonatiuh Sánchez-Vizuet, Manuel E. Solano

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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).

Original languageEnglish (US)
Pages (from-to)120-132
Number of pages13
JournalComputer Physics Communications
StatePublished - Feb 2019
Externally publishedYes


  • Anderson acceleration
  • Curved boundary
  • Grad–Shafranov
  • Hybridizable Discontinuous Galerkin (HDG)
  • Magnetohydrodynamics (MHD)
  • Plasma equilibrium

ASJC Scopus subject areas

  • Hardware and Architecture
  • Physics and Astronomy(all)


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