A Parallel Cut-Cell Algorithm for the Free-Boundary Grad--Shafranov Problem

Liu, Shuang; Tang, Qi; Tang, Xian-zhu

doi:10.1137/20m1385470

Title: A Parallel Cut-Cell Algorithm for the Free-Boundary Grad--Shafranov Problem

Journal Article · 13 December 2021 · SIAM Journal on Scientific Computing

DOI: https://doi.org/10.1137/20m1385470 · OSTI ID:1864994

Liu, Shuang ^[1];

^[2];

^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Univ. of California, San Diego, La Jolla, CA (United States)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

A parallel cut-cell algorithm is described to solve the free-boundary problem of the Grad--Shafranov equation. The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's acceleration for the resulting nonlinear problem, and a Cartesian grid embedded boundary method to handle the complex geometry. Here the algorithm is implemented in parallel using a standard domain-decomposition approach and a good parallel scaling is observed. Numerical results verify the accuracy and efficiency of the free-boundary Grad--Shafranov solver.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: Tokamak Disruption Simulation (TDS); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1864994

Report Number(s):: LA-UR-20-29880

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 6 Vol. 43; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (44)

Computation of ideal MHD equilibria Lackner, K. Computer Physics Communications, Vol. 12, Issue 1 https://doi.org/10.1016/0010-4655(76)90008-4	journal	September 1976
Dynamic modeling of transport and positional control of tokamaks Jardin, S. C.; Pomphrey, N.; Delucia, J. Journal of Computational Physics, Vol. 66, Issue 2 https://doi.org/10.1016/0021-9991(86)90077-x	journal	October 1986
Rational functions for guaranteed and experimentally well-conditioned global interpolation Berrut, J. -P. Computers & Mathematics with Applications, Vol. 15, Issue 1 https://doi.org/10.1016/0898-1221(88)90067-3	journal	January 1988
Overview of the SPARC tokamak Creely, A. J.; Greenwald, M. J.; Ballinger, S. B. Journal of Plasma Physics, Vol. 86, Issue 5 https://doi.org/10.1017/s0022377820001257	journal	September 2020
A higher-order finite-volume discretization method for Poisson's equation in cut cell geometries Devendran, D.; Graves, D. T.; Johansen, H. arXiv https://doi.org/10.48550/arxiv.1411.4283	preprint	January 2014
A Hybridizable Discontinuous Galerkin solver for the Grad-Shafranov equation Sánchez-Vizuet, Tonatiuh; Solano, Manuel arXiv https://doi.org/10.48550/arxiv.1712.04148	text	January 2017
Dimits shift, avalanche-like bursts, and Solitary propagating structures in the two-field Flux-Balanced Hasegawa-Wakatani model for plasma edge turbulence Qi, Di; Majda, Andrew J.; Cerfon, Antoine J. arXiv https://doi.org/10.48550/arxiv.2006.10554	text	January 2020
A Simple Level Set Method for Solving Stefan Problems Chen, S.; Merriman, B.; Osher, S. Journal of Computational Physics, Vol. 135, Issue 1 https://doi.org/10.1006/jcph.1997.5721	journal	July 1997
A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains Johansen, Hans; Colella, Phillip Journal of Computational Physics, Vol. 147, Issue 1 https://doi.org/10.1006/jcph.1998.5965	journal	November 1998
A Second-Order-Accurate Symmetric Discretization of the Poisson Equation on Irregular Domains Gibou, Frederic; Fedkiw, Ronald P.; Cheng, Li-Tien Journal of Computational Physics, Vol. 176, Issue 1 https://doi.org/10.1006/jcph.2001.6977	journal	February 2002
Barycentric rational interpolation with no poles and high rates of approximation Floater, Michael S.; Hormann, Kai Numerische Mathematik, Vol. 107, Issue 2 https://doi.org/10.1007/s00211-007-0093-y	journal	June 2007
Computation of ideal MHD equilibria Lackner, K. Computer Physics Communications, Vol. 12, Issue 1 https://doi.org/10.1016/0010-4655(76)90008-4	journal	September 1976
The CHEASE code for toroidal MHD equilibria Lütjens, H.; Bondeson, A.; Sauter, O. Computer Physics Communications, Vol. 97, Issue 3 https://doi.org/10.1016/0010-4655(96)00046-X	journal	September 1996
Numerical determination of axisymmetric toroidal magnetohydrodynamic equilibria Johnson, J. L.; Dalhed, H. E.; Greene, J. M. Journal of Computational Physics, Vol. 32, Issue 2 https://doi.org/10.1016/0021-9991(79)90129-3	journal	August 1979
Dynamic modeling of transport and positional control of tokamaks Jardin, S. C.; Pomphrey, N.; Delucia, J. Journal of Computational Physics, Vol. 66, Issue 2 https://doi.org/10.1016/0021-9991(86)90077-X	journal	October 1986
Remarks on the Grad-Shafranov equation Ambrosetti, A.; Calahorrano, M.; Dobarro, F. Applied Mathematics Letters, Vol. 3, Issue 3 https://doi.org/10.1016/0893-9659(90)90126-V	journal	January 1990
Rational functions for guaranteed and experimentally well-conditioned global interpolation Berrut, J. -P. Computers & Mathematics with Applications, Vol. 15, Issue 1 https://doi.org/10.1016/0898-1221(88)90067-3	journal	January 1988
Recent advances in linear barycentric rational interpolation Berrut, Jean-Paul; Klein, Georges Journal of Computational and Applied Mathematics, Vol. 259 https://doi.org/10.1016/j.cam.2013.03.044	journal	March 2014
Simulation technique of free-boundary equilibrium evolution in plasma ramp-up phase Honda, M. Computer Physics Communications, Vol. 181, Issue 9 https://doi.org/10.1016/j.cpc.2010.04.014	journal	September 2010
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
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 Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation Sánchez-Vizuet, Tonatiuh; Solano, Manuel E. Computer Physics Communications, Vol. 235 https://doi.org/10.1016/j.cpc.2018.09.013	journal	February 2019
Solving the Grad–Shafranov equation using spectral elements for tokamak equilibrium with toroidal rotation Li, Haolong; Zhu, Ping Computer Physics Communications, Vol. 260 https://doi.org/10.1016/j.cpc.2020.107264	journal	March 2021
The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions Jomaa, Z.; Macaskill, C. Journal of Computational Physics, Vol. 202, Issue 2 https://doi.org/10.1016/j.jcp.2004.07.011	journal	January 2005
A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem Gibou, Frédéric; Fedkiw, Ronald Journal of Computational Physics, Vol. 202, Issue 2 https://doi.org/10.1016/j.jcp.2004.07.018	journal	January 2005
A Cartesian grid embedded boundary method for the heat equation and Poisson’s equation in three dimensions Schwartz, Peter; Barad, Michael; Colella, Phillip Journal of Computational Physics, Vol. 211, Issue 2 https://doi.org/10.1016/j.jcp.2005.06.010	journal	January 2006
Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies Borazjani, Iman; Ge, Liang; Sotiropoulos, Fotis Journal of Computational Physics, Vol. 227, Issue 16 https://doi.org/10.1016/j.jcp.2008.04.028	journal	August 2008
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
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
FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains Faugeras, Blaise; Heumann, Holger Journal of Computational Physics, Vol. 343 https://doi.org/10.1016/j.jcp.2017.04.047	journal	August 2017
A stable partitioned FSI algorithm for rigid bodies and incompressible flow. Part II: General formulation Banks, J. W.; Henshaw, W. D.; Schwendeman, D. W. Journal of Computational Physics, Vol. 343 https://doi.org/10.1016/j.jcp.2017.04.064	journal	August 2017
A stable partitioned FSI algorithm for rigid bodies and incompressible flow in three dimensions Banks, J. W.; Henshaw, W. D.; Schwendeman, D. W. Journal of Computational Physics, Vol. 373 https://doi.org/10.1016/j.jcp.2018.06.072	journal	November 2018
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
Overview of the SPARC tokamak Creely, A. J.; Greenwald, M. J.; Ballinger, S. B. Journal of Plasma Physics, Vol. 86, Issue 5 https://doi.org/10.1017/S0022377820001257	journal	September 2020
Numerical studies of a class of reaction–diffusion equations with Stefan conditions Liu, Shuang; Du, Yihong; Liu, Xinfeng International Journal of Computer Mathematics, Vol. 97, Issue 5 https://doi.org/10.1080/00207160.2019.1599868	journal	April 2019
Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter Golub, Gene H.; Heath, Michael; Wahba, Grace Technometrics, Vol. 21, Issue 2 https://doi.org/10.1080/00401706.1979.10489751	journal	May 1979
Modelling toroidal rotation damping in ITER due to external 3D fields Liu, Yueqiang; Akers, R.; Chapman, I. T. Nuclear Fusion, Vol. 55, Issue 6 https://doi.org/10.1088/0029-5515/55/6/063027	journal	May 2015
Some new aspects of rational interpolation Schneider, Claus; Werner, Wilhelm Mathematics of Computation, Vol. 47, Issue 175 https://doi.org/10.1090/S0025-5718-1986-0842136-8	journal	January 1986
The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources LeVeque, Randall J.; Li, Zhilin SIAM Journal on Numerical Analysis, Vol. 31, Issue 4, p. 1019-1044 https://doi.org/10.1137/0731054	journal	August 1994
An Adaptive Discontinuous Petrov--Galerkin Method for the Grad--Shafranov Equation Peng, Zhichao; Tang, Qi; Tang, Xian-Zhu SIAM Journal on Scientific Computing, Vol. 42, Issue 5 https://doi.org/10.1137/19M1309894	journal	January 2020
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
A fourth-order Cartesian grid embedded boundary method for Poisson’s equation Devendran, Dharshi; Graves, Daniel; Johansen, Hans Communications in Applied Mathematics and Computational Science, Vol. 12, Issue 1 https://doi.org/10.2140/camcos.2017.12.51	journal	January 2017
Numerical Methods for a Two-Species Competition-Diffusion Model with Free Boundaries Liu, Shuang; Liu, Xinfeng Mathematics, Vol. 6, Issue 5 https://doi.org/10.3390/math6050072	journal	May 2018
Development of a free-boundary tokamak equilibrium solver for advanced study of tokamak equilibria Jeon, Young Mu Journal of the Korean Physical Society, Vol. 67, Issue 5 https://doi.org/10.3938/jkps.67.843	journal	September 2015

Similar Records

An Adaptive Newton-Based Free-Boundary Grad–Shafranov Solver

Journal Article · 2025 · SIAM Journal on Scientific Computing · OSTI ID:2999920

Serino, Daniel A.; Tang, Qi; Tang, Xian-Zhu; +2 more

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

Journal Article · 2020 · SIAM Journal on Scientific Computing · OSTI ID:1726182

Peng, Zhichao; Tang, Qi; Tang, Xian-Zhu

On the Grad--Shafranov equation as an eigenvalue problem, with implications for [ital q] solvers

Journal Article · 1994 · Physics of Plasmas; (United States) · OSTI ID:5184789

LoDestro, L L; Pearlstein, L D

Related Subjects

97 MATHEMATICS AND COMPUTING
Grad--Shafranov equation
cut-cell
finite difference
free-boundary problem

Title: A Parallel Cut-Cell Algorithm for the Free-Boundary Grad--Shafranov Problem

Citation Formats

References (44)

Similar Records

Related Subjects