A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Sánchez-Vizuet, Tonatiuh; Solano, Manuel E.

doi:10.1016/j.cpc.2018.09.013

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Journal Article · Fri Sep 28 00:00:00 EDT 2018 · Computer Physics Communications

DOI:https://doi.org/10.1016/j.cpc.2018.09.013· OSTI ID:1610077

^[1]; Solano, Manuel E. ^[2]

New York Univ. (NYU), NY (United States); DOE/OSTI
Univ. de Concepción (Chile)

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).

View Accepted Manuscript (DOE)

Research Organization:: New York Univ. (NYU), NY (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: FG02-86ER53233

OSTI ID:: 1610077

Alternate ID(s):: OSTI ID: 1635825

Journal Information:: Computer Physics Communications, Journal Name: Computer Physics Communications Vol. 235; ISSN 0010-4655

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (43)

Axisymmetric MHD equilibrium solver with bicubic Hermite elements Lütjens, H.; Bondeson, A.; Roy, A. Computer Physics Communications, Vol. 69, Issue 2-3 https://doi.org/10.1016/0010-4655(92)90167-w	journal	March 1992
Hydromagnetic equilibria and force-free fields Grad, Harold; Rubin, Hanan Journal of Nuclear Energy (1954), Vol. 7, Issue 3-4 https://doi.org/10.1016/0891-3919(58)90139-6	journal	September 1958
ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria Lee, Jungpyo; Cerfon, Antoine Computer Physics Communications, Vol. 190 https://doi.org/10.1016/j.cpc.2015.01.015	journal	May 2015
A mimetic spectral element solver for the Grad–Shafranov equation Palha, A.; Koren, B.; Felici, F. Journal of Computational Physics, Vol. 316 https://doi.org/10.1016/j.jcp.2016.04.002	journal	July 2016
A finite element method for interface problems in domains with smooth boundaries and interfaces Bramble, James H.; King, J. Thomas Advances in Computational Mathematics, Vol. 6, Issue 1 https://doi.org/10.1007/BF02127700	journal	December 1996
To CG or to HDG: A Comparative Study Kirby, Robert M.; Sherwin, Spencer J.; Cockburn, Bernardo Journal of Scientific Computing, Vol. 51, Issue 1 https://doi.org/10.1007/s10915-011-9501-7	journal	July 2011
To CG or to HDG: A Comparative Study in 3D Yakovlev, Sergey; Moxey, David; Kirby, Robert M. Journal of Scientific Computing, Vol. 67, Issue 1 https://doi.org/10.1007/s10915-015-0076-6	journal	July 2015
Axisymmetric MHD equilibrium solver with bicubic Hermite elements Lütjens, H.; Bondeson, A.; Roy, A. Computer Physics Communications, Vol. 69, Issue 2-3 https://doi.org/10.1016/0010-4655(92)90167-W	journal	March 1992
Flow patterns around heart valves: A numerical method Peskin, Charles S. Journal of Computational Physics, Vol. 10, Issue 2 https://doi.org/10.1016/0021-9991(72)90065-4	journal	October 1972
Computation of MHD equilibrium of tokamak plasma Takeda, Tatsuoki; Tokuda, Shinji Journal of Computational Physics, Vol. 93, Issue 1 https://doi.org/10.1016/0021-9991(91)90074-U	journal	March 1991
The ITER divertor concept Janeschitz, G.; Borrass, K.; Federici, G. Journal of Nuclear Materials, Vol. 220-222 https://doi.org/10.1016/0022-3115(94)00447-1	journal	April 1995
Engineering and design aspects related to the development of the ITER divertor Dietz, K. Fusion Engineering and Design, Vol. 27, Issue 1-2 https://doi.org/10.1016/0920-3796(94)00242-Y	journal	March 1995
Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions Nath, D.; Kalra, M. S.; Munshi, P. Computers & Mathematics with Applications, Vol. 70, Issue 5 https://doi.org/10.1016/j.camwa.2015.07.013	journal	September 2015
Solving the Grad–Shafranov equation with spectral elements Howell, E. C.; Sovinec, C. R. Computer Physics Communications, Vol. 185, Issue 5 https://doi.org/10.1016/j.cpc.2014.02.008	journal	May 2014
Numerical solution of ideal MHD equilibrium via radial basis functions collocation and moving least squares approximation methods Ghasemi, Maryam; Amrollahi, Reza Engineering Analysis with Boundary Elements, Vol. 67 https://doi.org/10.1016/j.enganabound.2016.02.008	journal	June 2016
High-resolution magnetohydrodynamic equilibrium code for unity beta plasmas Gourdain, P. -A.; Leboeuf, J. -N.; Neches, R. Y. Journal of Computational Physics, Vol. 216, Issue 1 https://doi.org/10.1016/j.jcp.2005.12.005	journal	July 2006
A fast, high-order solver for the Grad–Shafranov equation Pataki, Andras; Cerfon, Antoine J.; Freidberg, Jeffrey P. Journal of Computational Physics, Vol. 243 https://doi.org/10.1016/j.jcp.2013.02.045	journal	June 2013
Meshless method for solving fixed boundary problem of plasma equilibrium Imazawa, Ryota; Kawano, Yasunori; Itami, Kiyoshi Journal of Computational Physics, Vol. 292 https://doi.org/10.1016/j.jcp.2015.03.035	journal	July 2015
Accurate derivative evaluation for any Grad–Shafranov solver Ricketson, L. F.; Cerfon, A. J.; Rachh, M. Journal of Computational Physics, Vol. 305 https://doi.org/10.1016/j.jcp.2015.11.015	journal	January 2016
A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries Heumann, Holger; Rapetti, Francesca Journal of Computational Physics, Vol. 334 https://doi.org/10.1016/j.jcp.2017.01.006	journal	April 2017
Solution of Grad–Shafranov equation by the method of fundamental solutions Nath, D.; Kalra, M. S. Journal of Plasma Physics, Vol. 80, Issue 3 https://doi.org/10.1017/S0022377814000026	journal	February 2014
Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications Heumann, H.; Blum, J.; Boulbe, C. Journal of Plasma Physics, Vol. 81, Issue 3 https://doi.org/10.1017/S0022377814001251	journal	January 2015
A family of analytic equilibrium solutions for the Grad–Shafranov equation Guazzotto, L.; Freidberg, J. P. Physics of Plasmas, Vol. 14, Issue 11 https://doi.org/10.1063/1.2803759	journal	November 2007
“One size fits all” analytic solutions to the Grad–Shafranov equation Cerfon, Antoine J.; Freidberg, Jeffrey P. Physics of Plasmas, Vol. 17, Issue 3 https://doi.org/10.1063/1.3328818	journal	March 2010
Noncircular, finite aspect ratio, local equilibrium model Miller, R. L.; Chu, M. S.; Greene, J. M. Physics of Plasmas, Vol. 5, Issue 4 https://doi.org/10.1063/1.872666	journal	April 1998
Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions Mc Carthy, P. J. Physics of Plasmas, Vol. 6, Issue 9 https://doi.org/10.1063/1.873630	journal	September 1999
A projection-based error analysis of HDG methods Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Sayas, Francisco-Javier Mathematics of Computation, Vol. 79, Issue 271 https://doi.org/10.1090/S0025-5718-10-02334-3	journal	March 2010
Conditions for superconvergence of HDG methods for second-order elliptic problems Cockburn, Bernardo; Qiu, Weifeng; Shi, Ke Mathematics of Computation, Vol. 81, Issue 279 https://doi.org/10.1090/S0025-5718-2011-02550-0	journal	September 2012
A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity Cockburn, Bernardo; Qiu, Weifeng; Solano, Manuel Mathematics of Computation, Vol. 83, Issue 286 https://doi.org/10.1090/S0025-5718-2013-02747-0	journal	July 2013
Application of Parallelized Multigrid Method to Solution of MHD Equilibrium With MPI Hanawa, T.; Ikuno, S.; Kamitani, A. IEEE Transactions on Magnetics, Vol. 40, Issue 2 https://doi.org/10.1109/TMAG.2004.825440	journal	March 2004
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho SIAM Journal on Numerical Analysis, Vol. 47, Issue 2 https://doi.org/10.1137/070706616	journal	January 2009
Optimal Isoparametric Finite Elements and Error Estimates for Domains Involving Curved Boundaries Lenoir, M. SIAM Journal on Numerical Analysis, Vol. 23, Issue 3 https://doi.org/10.1137/0723036	journal	June 1986
A Novel Two-Grid Method for Semilinear Elliptic Equations Xu, Jinchao SIAM Journal on Scientific Computing, Vol. 15, Issue 1 https://doi.org/10.1137/0915016	journal	January 1994
Anderson Acceleration for Fixed-Point Iterations Walker, Homer F.; Ni, Peng SIAM Journal on Numerical Analysis, Vol. 49, Issue 4 https://doi.org/10.1137/10078356X	journal	January 2011
Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains Cockburn, Bernardo; Solano, Manuel SIAM Journal on Scientific Computing, Vol. 34, Issue 1 https://doi.org/10.1137/100805200	journal	January 2012
Convergence Analysis for Anderson Acceleration Toth, Alex; Kelley, C. T. SIAM Journal on Numerical Analysis, Vol. 53, Issue 2 https://doi.org/10.1137/130919398	journal	January 2015
A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems Cockburn, Bernardo; Gopalakrishnan, Jayadeep SIAM Journal on Numerical Analysis, Vol. 42, Issue 1 https://doi.org/10.1137/S0036142902417893	journal	January 2004
Two-Grid Discretization Techniques for Linear and Nonlinear PDE s Xu, Jinchao SIAM Journal on Numerical Analysis, Vol. 33, Issue 5 https://doi.org/10.1137/S0036142992232949	journal	October 1996
Isoparametric Bicubic Hermite Elements for Solution of the Grad-Shafranov Equation Huysmans, G. T. A.; Goedbloed, J. P.; Kerner, W. International Journal of Modern Physics C, Vol. 02, Issue 01 https://doi.org/10.1142/S0129183191000512	journal	March 1991
Algorithm 949: MATLAB Tools for HDG in Three Dimensions Fu, Zhixing; Gatica, Luis F.; Sayas, Francisco-javier ACM Transactions on Mathematical Software, Vol. 41, Issue 3 https://doi.org/10.1145/2658992	journal	June 2015
Iterative Procedures for Nonlinear Integral Equations Anderson, Donald G. Journal of the ACM, Vol. 12, Issue 4 https://doi.org/10.1145/321296.321305	journal	October 1965
Axialsymmetrische magnetohydrodynamische Gleichgewichtskonfigurationen Lüst, R.; Schlüter, A. Zeitschrift für Naturforschung A, Vol. 12, Issue 10 https://doi.org/10.1515/zna-1957-1014	journal	October 1957
Reduction of stiffness and mass matrices Guyan, Robert J. AIAA Journal, Vol. 3, Issue 2 https://doi.org/10.2514/3.2874	journal	February 1965

Cited By (3)

A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems Sánchez, Nestor; Sánchez-Vizuet, Tonatiuh; Solano, Manuel E. arXiv https://doi.org/10.48550/arxiv.1911.12298	preprint	January 2019
A parallel cut-cell algorithm for the free-boundary Grad-Shafranov problem Liu, Shuang; Tang, Qi; Tang, Xian-Zhu arXiv https://doi.org/10.48550/arxiv.2012.06015	preprint	January 2020
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity Giacomini, Matteo; Sevilla, Ruben arXiv https://doi.org/10.48550/arxiv.1908.06604	text	January 2019

Similar Records

Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains

Journal Article · Mon Mar 02 19:00:00 EST 2020 · Computer Physics Communications · OSTI ID:1849943

An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation

Journal Article · Wed Oct 07 20:00:00 EDT 2020 · SIAM Journal on Scientific Computing · OSTI ID:1726182

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY
Anderson acceleration
Grad–Shafranov
Hybridizable Discontinuous Galerkin (HDG)
curved boundary
magnetohydrodynamics (MHD)
plasma equilibrium

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Citation Formats

References (43)

Cited By (3)

Similar Records

Related Subjects