TY - JOUR

T1 - A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

AU - Sánchez-Vizuet, Tonatiuh

AU - Solano, Manuel E.

N1 - Funding Information:
The authors are deeply thankful to Antoine Cerfon for his numerous suggestions and remarks that contributed to greatly improve the quality of the work and to point out several possible paths for future and current improvements. The paper greatly benefited from the detailed comments and suggestions of the anonymous referees, for which the authors are grateful. Tonatiuh Sánchez-Vizuet was partially supported by the US Department of Energy . Grant No. DE-FG02-86ER53233 . Manuel Solano was partially supported by CONICYT – Chile through FONDECYT Project No. 1160320 , by BASAL Project CMM, Universidad de Chile and by Centro de Investigación en Ingeniería Matemática (CIMA), Universidad de Concepción.
Funding Information:
The authors are deeply thankful to Antoine Cerfon for his numerous suggestions and remarks that contributed to greatly improve the quality of the work and to point out several possible paths for future and current improvements. The paper greatly benefited from the detailed comments and suggestions of the anonymous referees, for which the authors are grateful. Tonatiuh Sánchez-Vizuet was partially supported by the US Department of Energy. Grant No. DE-FG02-86ER53233. Manuel Solano was partially supported by CONICYT – Chile through FONDECYT Project No. 1160320, by BASAL Project CMM, Universidad de Chile and by Centro de Investigación en Ingeniería Matemática (CI
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2019/2

Y1 - 2019/2

N2 - 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).

AB - 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).

KW - Anderson acceleration

KW - Curved boundary

KW - Grad–Shafranov

KW - Hybridizable Discontinuous Galerkin (HDG)

KW - Magnetohydrodynamics (MHD)

KW - Plasma equilibrium

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U2 - 10.1016/j.cpc.2018.09.013

DO - 10.1016/j.cpc.2018.09.013

M3 - Article

AN - SCOPUS:85054744407

SN - 0010-4655

VL - 235

SP - 120

EP - 132

JO - Computer Physics Communications

JF - Computer Physics Communications

ER -