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Title: A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator

Journal Article · · IMA Journal of Numerical Analysis
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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.

Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
AC52-06NA25396
OSTI ID:
1304700
Report Number(s):
LA-UR-14-21207
Journal Information:
IMA Journal of Numerical Analysis, Vol. 36, Issue 2; ISSN 0272-4979
Publisher:
Oxford University Press/Institute of Mathematics and its ApplicationsCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 14 works
Citation information provided by
Web of Science

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Cited By (4)

A parallel time integrator for solving the linearized shallow water equations on the rotating sphere: Parallel time integrator for linearized SWE on rotating sphere journal October 2018
A Parallel Time-Integrator for Solving the Linearized Shallow Water Equations on the Rotating Sphere text January 2018
Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere journal April 2020
An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs journal July 2021

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Related Subjects

97 MATHEMATICS AND COMPUTING
time-stepping methods
optimal rational approximations
parallel-in-time
direct solver