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Journal Articles SMAI Journal of Computational Mathematics Year : 2021

Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes

Abstract

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.
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Dates and versions

hal-02894841 , version 1 (09-07-2020)
hal-02894841 , version 2 (16-11-2021)

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Joao Guilherme Caldas Steinstraesser, Vincent Guinot, Antoine Rousseau. Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes. SMAI Journal of Computational Mathematics, 2021, 7, pp.159-184. ⟨10.5802/smai-jcm.75⟩. ⟨hal-02894841v2⟩
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