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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Vydáno v:	Numerical algorithms Ročník 86; číslo 4; s. 1685 - 1703
Hlavní autoři:	Song, Bo, Jiang, Yao-Lin, Wang, Xiaolong
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.04.2021 Springer Nature B.V
Témata:	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
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Abstract	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.
AbstractList	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.
Author	Wang, Xiaolong Song, Bo Jiang, Yao-Lin
Author_xml	– sequence: 1 givenname: Bo orcidid: 0000-0001-7066-6949 surname: Song fullname: Song, Bo email: bosong@nwpu.edu.cn organization: School of Mathematics and Statistics, Northwestern Polytechnical University – sequence: 2 givenname: Yao-Lin surname: Jiang fullname: Jiang, Yao-Lin organization: School of Mathematics and Statistics, Xi’an Jiaotong University – sequence: 3 givenname: Xiaolong surname: Wang fullname: Wang, Xiaolong organization: School of Mathematics and Statistics, Northwestern Polytechnical University
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SubjectTerms	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
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Title	Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation
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