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Published in:
Robust Block Preconditioners for Biot’s Model Read chapter Read first chapter

2018 | OriginalPaper | Chapter

A Nonlinear ParaExp Algorithm

Authors : Martin J. Gander, Stefan Güttel, Madalina Petcu

Published in: Domain Decomposition Methods in Science and Engineering XXIV

Publisher: Springer International Publishing

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Abstract

We derive and analyze a nonlinear variant of the ParaExp algorithm introduced in Gander and Güttel (SIAM J Sci Comput 35(2):C123–C142, 2013) for linear evolution problems. We show that the nonlinear ParaExp algorithm converges in a finite number of steps, and that it can be interpreted as a parareal algorithm where the coarse integrator solves the linear part of the evolution problem. We also provide a numerical example illustrating the efficiency of the new algorithm.

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previous chapter On High-Order Approximation and Stability with Conservative Properties
next chapter On Optimal Coarse Spaces for Domain Decomposition and Their Approximation
Literature
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M.J. Gander, 50 years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition Methods (Springer, Berlin, 2015), pp. 69–113MATH
2.
M.J. Gander, S. Güttel, PARAEXP: a parallel integrator for linear initial-value problems. SIAM J. Sci. Comput. 35(2), C123–C142 (2013)MathSciNetCrossRef
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M.J. Gander, E. Hairer, Nonlinear convergence analysis for the parareal algorithm, in Domain Decomposition Methods in Science and Engineering XVII (Springer, Berlin, 2007), pp. 45–56
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M.J. Gander, L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing. Math. Comput. 74, 153–176 (2004)MathSciNetCrossRef
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M.J. Gander, S. Vanderwalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)MathSciNetCrossRef
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J. Gopalakrishnan, J. Schöberl, C. Wintersteiger, Mapped tent pitching schemes for hyperbolic systems. SIAM J. Sci. Comput. 39(6), B1043–B1063 (2017)MathSciNetCrossRef
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S. Güttel, A parallel overlapping time-domain decomposition method for ODEs, in Domain Decomposition Methods in Science and Engineering XX (Springer, Berlin, 2013), pp. 483–490
10.
G. Kooij, M. Botchev, B. Geurts, A block Krylov subspace implementation of the time-parallel ParaExp method and its extension for nonlinear partial differential equations. J. Comput. Appl. Math. 316, 229–246 (2017)MathSciNetCrossRef
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J.-L. Lions, Y. Maday, G. Turinici, A “parareal” in time discretization of PDE’s. C. R. Acad. Sci. Paris Sér. I Math. 332, 661–668 (2001)MathSciNetCrossRef
Metadata
Title
A Nonlinear ParaExp Algorithm
Authors
Martin J. Gander
Stefan Güttel
Madalina Petcu
Copyright Year
2018
Book
Domain Decomposition Methods in Science and Engineering XXIV

Print ISBN: 978-3-319-93872-1

Electronic ISBN: 978-3-319-93873-8

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-93873-8_24

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