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
CC-BY-NC-ND 4.0
@article{SMAI-JCM_2021__7__159_0,
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 = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.75/}
}
TY - JOUR AU - Joao G. Caldas Steinstraesser AU - Vincent Guinot AU - Antoine Rousseau 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 - https://smai-jcm.centre-mersenne.org/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 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 %V 7 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.75/ %R 10.5802/smai-jcm.75 %G en %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. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.75/
[1] , 55th AIAA Aerospace Sciences Meeting (2017) | DOI
[2] Multiscale coupling of finite element and lattice Boltzmann methods for time dependent problems (2012) (https://hal.archives-ouvertes.fr/hal-00746942)
[3] Scheduling of tasks in the parareal algorithm, Parallel Computing, Volume 37 (2011) no. 3, pp. 172-182 | DOI | MR | Zbl
[4] Symplectic multi-time step parareal algorithms applied to molecular dynamics (2009) (https://hal.archives-ouvertes.fr/hal-00358459, submitted to SIAM Journal of Scientific Computing)
[5] An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 9, pp. 667-672 | DOI | MR | Zbl
[6] Coupling large and small scale shallow water models with porosity in the presence of anisotropy, Ph. D. Thesis, Université de Montpellier (2021)
[7] Nonlinear Model Reduction via Discrete Empirical Interpolation, SIAM J. Sci. Comput., Volume 32 (2010) no. 5, pp. 2737-2764 | DOI | MR | Zbl
[8] On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method, Reduced Order Methods for Modeling and Computational Reduction, Springer, 2014, pp. 187-214 | DOI | Zbl
[9] Greedy Non-Intrusive Reduced Order Model for Fluid Dynamics, AIAA J., Volume 56 (2018), p. 12 | DOI
[10] Stable Parareal in Time Method for First- and Second-Order Hyperbolic Systems, SIAM J. Sci. Comput., Volume 35 (2013) no. 1, p. A52-A78 | DOI | MR | Zbl
[11] The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale, SIAM Rev., Volume 60 (2018) no. 4, pp. 808-865 | DOI | MR | Zbl
[12] Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies, ESAIM, Math. Model. Numer. Anal., Volume 45 (2011) no. 5, pp. 825-852 | DOI | Numdam | MR | Zbl
[13] Acceleration of unsteady hydrodynamic simulations using the parareal algorithm, J. Comput. Sci., Volume 19 (2017), pp. 57-76 | DOI
[14] Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses, Int. J. Numer. Methods Eng., Volume 67 (2006), pp. 697-724 | DOI | MR | Zbl
[15] , Domain Decomposition Methods in Science and Engineering (2005), pp. 433-440 | DOI | Zbl
[16] Analysis of the Parareal Algorithm Applied to Hyperbolic Problems Using Characteristics, Bol. Soc. Esp. Mat. Apl., Volume 42 (2008), pp. 21-35 https://archive-ouverte.unige.ch/unige:6268 (ID: unige:6268) | Zbl
[17] , Multiple Shooting and Time Domain Decomposition Methods (2015), pp. 69-113 | DOI | Zbl
[18] Analysis of a Krylov subspace enhanced parareal algorithm for linear problems, ESAIM, Proc., Volume 25 (2008), pp. 114-129 | DOI | MR | Zbl
[19] Flux and source term discretization in two-dimensional shallow water models with porosity on unstructured grids, Int. J. Numer. Methods Fluids, Volume 50 (2006) no. 3, pp. 309-345 | DOI | MR | Zbl
[20] MPI–OpenMP algorithms for the parallel space-time solution of time dependent PDEs, Domain decomposition methods in science and engineering XXI (Lecture Notes Computational Sciences and Engineering), Volume 98, Springer, 2014, pp. 179-187 | MR | Zbl
[21] Influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs, Comput. Vis. Sci., Volume 19 (2018) no. 3-4, pp. 97-108 | DOI | MR
[22] Résolution d’EDP par un schéma en temps ‘pararéel’, C. R. Math. Acad. Sci. Paris, Volume 332 (2001) no. 7, pp. 661-668 | DOI | Zbl
[23] The ‘Parareal in Time’ Algorithm, Substructuring Techniques and Domain Decomposition Methods (Computational science, engineering and technology series), Saxe-Coburg Publications, 2010, pp. 19-44 | DOI
[24] On the POD method: an abstract investigation with applications to reduced-order modeling and suboptimal control, Ph. D. Thesis, Georg-August-Universität (2008)
[25] Applications of time parallelization, Comput. Vis. Sci., Volume 23 (2020) no. 1-4, 11 | DOI | MR
[26] The parareal algorithm for American options, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 11, pp. 1132-1138 | DOI | MR | Zbl
[27] Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcation Chaos Appl. Sci. Eng., Volume 15 (2005) no. 3, pp. 997-1013 | DOI | MR | Zbl
[28] Wave propagation characteristics of Parareal, Comput. Vis. Sci., Volume 19 (2018), pp. 1-17 | DOI | MR | Zbl
[29] Explicit parallel-in-time integration of a linear acoustic-advection system, Computers Fluids, Volume 59 (2012), pp. 72-83 | DOI | MR | Zbl
[30] Time domain parallelization for computational geodynamics, Geochemistry, Geophysics, Geosystems, Volume 13 (2012) no. 1 | DOI
[32] Parallel-in-time simulation of the unsteady Navier–Stokes equations for incompressible flow, Int. J. Numer. Methods Fluids, Volume 45 (2004) no. 10, pp. 1123-1136 | DOI | Zbl
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