Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation

The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are nonoverlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of these space and time parallel strategies, we present and a...

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Bibliographic Details
Published in:Numerical algorithms Vol. 86; no. 4; pp. 1685 - 1703
Main Authors: Song, Bo, Jiang, Yao-Lin, Wang, Xiaolong
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2021
Springer Nature B.V
Subjects:
Algebra
Algorithms
Approximation
Boundary conditions
Computer Science
Decomposition
Dirichlet problem
Domain decomposition methods
Methods
Numeric Computing
Numerical Analysis
Ordinary differential equations
Original Paper
Partial differential equations
Relaxation method (mathematics)
Spacetime
Theory of Computation
Thermodynamics
Waveforms
Dirichlet-Neumann method
Waveform relaxation
Domain decomposition
Parareal method
Neumann-Neumann method
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are nonoverlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of these space and time parallel strategies, we present and analyze two parareal algorithms based on the Dirichlet-Neumann and the Neumann-Neumann waveform relaxation method for the heat equation by choosing Dirichlet-Neumann/Neumann-Neumann waveform relaxation as two new kinds of fine propagators instead of the classical fine propagator. Both new proposed algorithms could be viewed as a space-time parallel algorithm, which increases the parallelism both in space and in time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments finally.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-020-00949-y