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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.
Keywords: Parareal method, POD-DEIM, reduced-order model, finite volume, shallow water equations
Caldas Steinstraesser, Joao G. 1 ; Guinot, Vincent 2 ; Rousseau, Antoine 1
Caldas Steinstraesser, Joao G.; Guinot, Vincent; Rousseau, Antoine. 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
@article{SMAI-JCM_2021__7__159_0,
author = {Caldas Steinstraesser, Joao G. and Guinot, Vincent and Rousseau, Antoine},
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},
year = {2021},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {7},
doi = {10.5802/smai-jcm.75},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.75/}
}
TY - JOUR AU - Caldas Steinstraesser, Joao G. AU - Guinot, Vincent AU - Rousseau, Antoine TI - Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes JO - The SMAI Journal of computational mathematics PY - 2021 SP - 159 EP - 184 VL - 7 PB - Société de Mathématiques Appliquées et Industrielles UR - http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.75/ DO - 10.5802/smai-jcm.75 LA - en ID - SMAI-JCM_2021__7__159_0 ER -
%0 Journal Article %A Caldas Steinstraesser, Joao G. %A Guinot, Vincent %A Rousseau, Antoine %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 %V 7 %I Société de Mathématiques Appliquées et Industrielles %U http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.75/ %R 10.5802/smai-jcm.75 %G en %F SMAI-JCM_2021__7__159_0
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