Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes
The SMAI journal of computational mathematics, Volume 7 (2021), pp. 159-184.

In this work, the POD-DEIM-based parareal method introduced in [8] is implemented for solving the two-dimensional nonlinear shallow water equations using a finite volume scheme. This method is a variant of the traditional parareal method, first introduced by [22], that improves the stability and convergence for nonlinear hyperbolic problems, and uses reduced-order models constructed via the Proper Orthogonal Decomposition - Discrete Empirical Interpolation Method (POD-DEIM) applied to snapshots of the solution of the parareal iterations. We propose a modification of this parareal method for further stability and convergence improvements. It consists in enriching the snapshots set for the POD-DEIM procedure with extra snapshots whose computation does not require any additional computational cost. The performances of the classical parareal method, the POD-DEIM-based parareal method and our proposed modification are compared using numerical tests with increasing complexity. Our modified method shows a more stable behaviour and converges in fewer iterations than the other two methods.

Published online:
Classification: 68W10,  76B07
Keywords: Parareal method, POD-DEIM, reduced-order model, finite volume, shallow water equations
Joao G. Caldas Steinstraesser 1; Vincent Guinot 2; Antoine Rousseau 1

1. Inria, IMAG, Univ Montpellier, CNRS, Montpellier, France
2. Univ Montpellier, HSM, CNRS, IRD, Inria, Montpellier, France
     author = {Joao G. Caldas Steinstraesser and Vincent Guinot and Antoine Rousseau},
     title = {Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes},
     journal = {The SMAI journal of computational mathematics},
     pages = {159--184},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {7},
     year = {2021},
     doi = {10.5802/smai-jcm.75},
     language = {en},
     url = {}
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PY  - 2021
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PB  - Société de Mathématiques Appliquées et Industrielles
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%A Joao G. Caldas Steinstraesser
%A Vincent Guinot
%A Antoine Rousseau
%T Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes
%J The SMAI journal of computational mathematics
%D 2021
%P 159-184
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%I Société de Mathématiques Appliquées et Industrielles
%R 10.5802/smai-jcm.75
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%F SMAI-JCM_2021__7__159_0
Joao G. Caldas Steinstraesser; Vincent Guinot; Antoine Rousseau. Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes. The SMAI journal of computational mathematics, Volume 7 (2021), pp. 159-184. doi : 10.5802/smai-jcm.75.

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