A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator
Abstract
Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.
- Authors:
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1304700
- Report Number(s):
- LA-UR-14-21207
Journal ID: ISSN 0272-4979
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- IMA Journal of Numerical Analysis
- Additional Journal Information:
- Journal Volume: 36; Journal Issue: 2; Journal ID: ISSN 0272-4979
- Publisher:
- Oxford University Press/Institute of Mathematics and its Applications
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; time-stepping methods; optimal rational approximations; parallel-in-time; direct solver
Citation Formats
Haut, T. S., Babb, T., Martinsson, P. G., and Wingate, B. A. A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator. United States: N. p., 2015.
Web. doi:10.1093/imanum/drv021.
Haut, T. S., Babb, T., Martinsson, P. G., & Wingate, B. A. A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator. United States. https://doi.org/10.1093/imanum/drv021
Haut, T. S., Babb, T., Martinsson, P. G., and Wingate, B. A. Tue .
"A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator". United States. https://doi.org/10.1093/imanum/drv021. https://www.osti.gov/servlets/purl/1304700.
@article{osti_1304700,
title = {A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator},
author = {Haut, T. S. and Babb, T. and Martinsson, P. G. and Wingate, B. A.},
abstractNote = {Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.},
doi = {10.1093/imanum/drv021},
journal = {IMA Journal of Numerical Analysis},
number = 2,
volume = 36,
place = {United States},
year = {Tue Jun 16 00:00:00 EDT 2015},
month = {Tue Jun 16 00:00:00 EDT 2015}
}
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journal, October 2018
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- Numerical Linear Algebra with Applications, Vol. 26, Issue 2
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text, January 2018
- Schreiber, Martin; Loft, Richard
- arXiv
A parallel time integrator for solving the linearized shallow water equations on the rotating sphere: Parallel time integrator for linearized SWE on rotating sphere
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- Schreiber, Martin; Loft, Richard
- Numerical Linear Algebra with Applications, Vol. 26, Issue 2
Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere
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- Hamon, François P.; Schreiber, Martin; Minion, Michael L.
- Journal of Computational Physics, Vol. 407
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