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Solver comparison for Poisson-like equations on tokamak geometries

Bourne, Emily and Leleux, Philippe and Kormann, Katharina and Kruse, Carola and Grandgirard, Virginie and Güclü, Yaman and Kühn, Martin Joachim and Rüde, Ulrich and Sonnendrücker, Eric and Zoni, Edoardo (2023) Solver comparison for Poisson-like equations on tokamak geometries. Journal of Computational Physics, 488 (1). Elsevier. doi: 10.1016/j.jcp.2023.112249. ISSN 0021-9991.

[img] PDF - Preprint version (submitted draft)
[img] PDF - Only accessible within DLR bis 30 May 2025 - Postprint version (accepted manuscript)

Official URL: https://www.sciencedirect.com/science/article/abs/pii/S0021999123003443


The solution of Poisson-like equations defined on a complex geometry is required for gyrokinetic simulations, which are important for the modelling of plasma turbulence in nuclear fusion devices such as the ITER tokamak. In this paper, we compare three existing solvers finely tuned to solve this problem, in terms of the accuracy of the solution, and their computational efficiency. We also consider practical implementation aspects, including the parallel efficiency of the code, potentially enabling an integration of the solvers in a state-of-the-art first-principle gyrokinetic simulation framework. The first, the Spline FEM solver, uses C1 polar splines to construct a finite elements method which solves the equation on curvilinear coordinates. The resulting linear system is solved using a conjugate gradient method. The second, the GMGPolar solver, uses a symmetric finite difference method to discretise the differential equation. The resulting linear system is solved using a tailored geometric multigrid scheme, with a combination of zebra circle and radial line smoothers, together with an implicit extrapolation scheme. The third, the Embedded Boundary solver, uses a finite volumes method on Cartesian coordinates with an embedded boundary scheme. The resulting linear system is solved using a multigrid scheme. The Spline FEM solver is shown to be the most accurate. The GMGPolar solver is shown to use the least memory. The Embedded Boundary solver is shown to be the fastest in most cases. All three solvers are shown to be capable of solving the equation on a realistic non-analytical geometry. The Embedded Boundary solver is additionally used to attempt to solve an X-point geometry.

Item URL in elib:https://elib.dlr.de/195446/
Document Type:Article
Title:Solver comparison for Poisson-like equations on tokamak geometries
AuthorsInstitution or Email of AuthorsAuthor's ORCID iDORCID Put Code
Bourne, EmilyCEA, IRFM, Saint-Paul-les-Durance, F-13108, Francehttps://orcid.org/0000-0002-3469-2338UNSPECIFIED
Leleux, PhilippeUNSPECIFIEDhttps://orcid.org/0000-0002-3760-4698UNSPECIFIED
Kormann, KatharinaRuhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 150, 44801 Bochum, Germanyhttps://orcid.org/0000-0003-1956-2073UNSPECIFIED
Kruse, CarolaParallel Algorithms Team, CERFACS (Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique), 42 Avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, Francehttps://orcid.org/0000-0002-4142-7356UNSPECIFIED
Grandgirard, VirginieCEA, IRFM, Saint-Paul-les-Durance, F-13108, Francehttps://orcid.org/0000-0001-7821-9107UNSPECIFIED
Güclü, YamanMax-Planck Institut für Plasmaphysik, Garching, Germanyhttps://orcid.org/0000-0003-2619-5152UNSPECIFIED
Kühn, Martin JoachimUNSPECIFIEDhttps://orcid.org/0000-0002-0906-6984UNSPECIFIED
Rüde, UlrichLehrstuhl für Informatik 10 (Systemsimulation), Universität Erlangen-Nürnberg, Nürnberghttps://orcid.org/0000-0001-8796-8599UNSPECIFIED
Sonnendrücker, EricMax-Planck Institut für Plasmaphysik, Garching, Germanyhttps://orcid.org/0000-0002-8340-7230UNSPECIFIED
Zoni, EdoardoLawrence Berkeley National Laboratory, Berkeley, CA, USAhttps://orcid.org/0000-0001-5662-4646UNSPECIFIED
Date:September 2023
Journal or Publication Title:Journal of Computational Physics
Refereed publication:Yes
Open Access:No
Gold Open Access:No
In ISI Web of Science:Yes
Keywords:plasma simulation, Poisson equation, tokamak, finite elements, finite differences, finite volumes, multigrid scheme, conjugate gradient
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Space
HGF - Program Themes:Space System Technology
DLR - Research area:Raumfahrt
DLR - Program:R SY - Space System Technology
DLR - Research theme (Project):R - Tasks SISTEC
Location: Köln-Porz
Institutes and Institutions:Institute for Software Technology
Institute for Software Technology > High-Performance Computing
Deposited By: Kühn, Dr. Martin Joachim
Deposited On:26 Jun 2023 13:06
Last Modified:28 Jun 2023 14:17

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