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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
M.J. Gander, 50 years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition Methods (Springer, Berlin, 2015), pp. 69–113
M.J. Gander, S. Güttel, PARAEXP: a parallel integrator for linear initial-value problems. SIAM J. Sci. Comput. 35(2), C123–C142 (2013)
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
M.J. Gander, L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing. Math. Comput. 74, 153–176 (2004)
M.J. Gander, M. Petcu, Analysis of a Krylov subspace enhanced parareal algorithm. ESAIM Proc. 25, 45–56 (2008)
M.J. Gander, S. Vanderwalle, Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29(2), 556–578 (2007)
M.J. Gander, L. Halpern, F. Nataf, Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41, 1643–1681 (2003)
J. Gopalakrishnan, J. Schöberl, C. Wintersteiger, Mapped tent pitching schemes for hyperbolic systems. SIAM J. Sci. Comput. 39(6), B1043–B1063 (2017)
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
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)
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)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Gander, M.J., Güttel, S., Petcu, M. (2018). A Nonlinear ParaExp Algorithm. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-93873-8_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93872-1
Online ISBN: 978-3-319-93873-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)