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Title: On the performance of exponential integrators for problems in magnetohydrodynamics

Abstract

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EPIRK-type are used to simulate magnetic reconnection and the Kevin–Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDFmore » scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD.« less

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Univ. Innsbruck (Austria)
  2. Univ. of California, Merced, CA (United States)
  3. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
OSTI Identifier:
1860902
Alternate Identifier(s):
OSTI ID: 1398578
Report Number(s):
LLNL-JRNL-832030
Journal ID: ISSN 0021-9991; 1049472; TRN: US2305993
Grant/Contract Number:  
AC52-07NA27344; 1115978; P25346
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 330; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; exponential integrators; magnetohydrodynamics; stiff systems; EPIC

Citation Formats

Einkemmer, Lukas, Tokman, Mayya, and Loffeld, John. On the performance of exponential integrators for problems in magnetohydrodynamics. United States: N. p., 2016. Web. doi:10.1016/j.jcp.2016.11.027.
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Einkemmer, Lukas, Tokman, Mayya, & Loffeld, John. On the performance of exponential integrators for problems in magnetohydrodynamics. United States. https://doi.org/10.1016/j.jcp.2016.11.027
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Einkemmer, Lukas, Tokman, Mayya, and Loffeld, John. Sat . "On the performance of exponential integrators for problems in magnetohydrodynamics". United States. https://doi.org/10.1016/j.jcp.2016.11.027. https://www.osti.gov/servlets/purl/1860902.
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@article{osti_1860902,
title = {On the performance of exponential integrators for problems in magnetohydrodynamics},
author = {Einkemmer, Lukas and Tokman, Mayya and Loffeld, John},
abstractNote = {Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EPIRK-type are used to simulate magnetic reconnection and the Kevin–Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD.},
doi = {10.1016/j.jcp.2016.11.027},
journal = {Journal of Computational Physics},
number = ,
volume = 330,
place = {United States},
year = {Sat Dec 03 00:00:00 EST 2016},
month = {Sat Dec 03 00:00:00 EST 2016}
}
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