publication . Other literature type . Article . Preprint . 2019

Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

Martin Schreiber; Nathanaël Schaeffer; Richard Loft;
  • Published: 01 Jul 2019
  • Publisher: Elsevier BV
Abstract
Abstract High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approximation of exponential integrators (REXI). This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel. Previous numerical studies showed that this reformulation allows taking arbitrarily long t...
Subjects
free text keywords: Theoretical Computer Science, Computer Networks and Communications, Hardware and Architecture, Software, Artificial Intelligence, Computer Graphics and Computer-Aided Design, Applied mathematics, Spacetime, Integrator, Parallel computing, System of linear equations, Multiple time dimensions, Exponential integrator, Massively parallel, Shallow water equations, Computer science, Discretization, Computer Science - Numerical Analysis, Computer Science - Computational Engineering, Finance, and Science, [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], [INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC]
44 references, page 1 of 3

[1] H. Sutter, The free lunch is over: A fundamental turn toward concurrency in software, Dr. Dobb's journal 30 (3) (2005) 202{210.

[2] ECMWF, Ecmwf strategy 2016-2025, the strengh of a common goal (2016). [OpenAIRE]

[3] M. O ce, Met o ce science strategy: 2016-2021 delivering science with impact (2016).

[4] C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty- ve years later, SIAM review 45 (1) (2003) 3{49.

[5] M. Hochbruck, A. Ostermann, Exponential integrators, Acta Numerica 19.

[6] J. D. Lawson, Generalized runge-kutta processes for stable systems with large lipschitz constants, SIAM Journal on Numerical Analysis 4 (3) (1967) 372{380. [OpenAIRE]

[7] M. Hochbruck, C. Lubich, H. Selhofer, Exponential integrators for large systems of di erential equations, SIAM Journal on Scienti c Computing 19 (5).

[8] S. Cox, P. Matthews, Exponential time di erencing for sti systems, Journal of Computational Physics 176 (2) (2002) 430 { 455.

[9] M. Tokman, E cient integration of large sti systems of odes with exponential propagation iterative (epi) methods, Journal of Comp. Physics 213 (2).

[10] M. Tokman, A new class of exponential propagation iterative methods of runge{kutta type (epirk), Journal of Comp. Physics 230 (24) (2011) 8762{ 8778. [OpenAIRE]

[11] C. Clancy, P. Lynch, Laplace transform integration of the SWE. Part I: Eulerian form. and Kelvin waves, Quarterly Journal of the RMS 137 (656) (2011) 792{799.

[12] J. Niesen, W. M. Wright, Algorithm 919: A Krylov Subspace Algorithm for Evaluating the -Functions Appearing in Exponential Integrators, ACM Trans. Math. Softw. 38 (3) (2012) 22:1{22:19. doi:10.1145/2168773. 2168781. URL http://doi.acm.org/10.1145/2168773.2168781

[13] C. Clancy, J. A. Pudykiewicz, On the use of exponential time integration methods in atmospheric models, Tellus A 65 (2013) 1{16. [OpenAIRE]

[14] L. Bonaventura, Local exponential methods: a domain decomposition approach to exponential time integration of pdes, CoRR abs/1505.02248. URL http://arxiv.org/abs/1505.02248

[15] F. Garcia, L. Bonaventura, M. Net, J. Sanchez, Exp. versus IMEX highorder time int. for thermal conv. in rot. spherical shells, J. of Comp. Physics 264.

44 references, page 1 of 3
Abstract
Abstract High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approximation of exponential integrators (REXI). This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel. Previous numerical studies showed that this reformulation allows taking arbitrarily long t...
Subjects
free text keywords: Theoretical Computer Science, Computer Networks and Communications, Hardware and Architecture, Software, Artificial Intelligence, Computer Graphics and Computer-Aided Design, Applied mathematics, Spacetime, Integrator, Parallel computing, System of linear equations, Multiple time dimensions, Exponential integrator, Massively parallel, Shallow water equations, Computer science, Discretization, Computer Science - Numerical Analysis, Computer Science - Computational Engineering, Finance, and Science, [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], [INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC]
44 references, page 1 of 3

[1] H. Sutter, The free lunch is over: A fundamental turn toward concurrency in software, Dr. Dobb's journal 30 (3) (2005) 202{210.

[2] ECMWF, Ecmwf strategy 2016-2025, the strengh of a common goal (2016). [OpenAIRE]

[3] M. O ce, Met o ce science strategy: 2016-2021 delivering science with impact (2016).

[4] C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty- ve years later, SIAM review 45 (1) (2003) 3{49.

[5] M. Hochbruck, A. Ostermann, Exponential integrators, Acta Numerica 19.

[6] J. D. Lawson, Generalized runge-kutta processes for stable systems with large lipschitz constants, SIAM Journal on Numerical Analysis 4 (3) (1967) 372{380. [OpenAIRE]

[7] M. Hochbruck, C. Lubich, H. Selhofer, Exponential integrators for large systems of di erential equations, SIAM Journal on Scienti c Computing 19 (5).

[8] S. Cox, P. Matthews, Exponential time di erencing for sti systems, Journal of Computational Physics 176 (2) (2002) 430 { 455.

[9] M. Tokman, E cient integration of large sti systems of odes with exponential propagation iterative (epi) methods, Journal of Comp. Physics 213 (2).

[10] M. Tokman, A new class of exponential propagation iterative methods of runge{kutta type (epirk), Journal of Comp. Physics 230 (24) (2011) 8762{ 8778. [OpenAIRE]

[11] C. Clancy, P. Lynch, Laplace transform integration of the SWE. Part I: Eulerian form. and Kelvin waves, Quarterly Journal of the RMS 137 (656) (2011) 792{799.

[12] J. Niesen, W. M. Wright, Algorithm 919: A Krylov Subspace Algorithm for Evaluating the -Functions Appearing in Exponential Integrators, ACM Trans. Math. Softw. 38 (3) (2012) 22:1{22:19. doi:10.1145/2168773. 2168781. URL http://doi.acm.org/10.1145/2168773.2168781

[13] C. Clancy, J. A. Pudykiewicz, On the use of exponential time integration methods in atmospheric models, Tellus A 65 (2013) 1{16. [OpenAIRE]

[14] L. Bonaventura, Local exponential methods: a domain decomposition approach to exponential time integration of pdes, CoRR abs/1505.02248. URL http://arxiv.org/abs/1505.02248

[15] F. Garcia, L. Bonaventura, M. Net, J. Sanchez, Exp. versus IMEX highorder time int. for thermal conv. in rot. spherical shells, J. of Comp. Physics 264.

44 references, page 1 of 3
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publication . Other literature type . Article . Preprint . 2019

Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

Martin Schreiber; Nathanaël Schaeffer; Richard Loft;