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:
DOI: 10.5802/smai-jcm.75
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
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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] N. Akkari; R. Mercier; G. Lartigue; V. Moureau, 55th AIAA Aerospace Sciences Meeting (2017) | DOI

[2] M. Astorino; F. Chouly; A. Quarteroni Multiscale coupling of finite element and lattice Boltzmann methods for time dependent problems (2012) (https://hal.archives-ouvertes.fr/hal-00746942)

[3] E. Aubanel Scheduling of tasks in the parareal algorithm, Parallel Computing, Volume 37 (2011) no. 3, pp. 172-182 | DOI | MR | Zbl

[4] C. Audouze; M. Massot; S. Volz 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] M. Barrault; Y. Maday; N. C. Nguyen; A. T. Patera 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] J. G. Caldas Steinstraesser Coupling large and small scale shallow water models with porosity in the presence of anisotropy, Ph. D. Thesis, Université de Montpellier (2021)

[7] S. Chaturantabut; D. C. Sorensen Nonlinear Model Reduction via Discrete Empirical Interpolation, SIAM J. Sci. Comput., Volume 32 (2010) no. 5, pp. 2737-2764 | DOI | MR | Zbl

[8] F. Chen; J. S. Hesthaven; X. Zhu 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] W. Chen; J. S. Hesthaven; B. Junqiang; Y. Qiu; Y. Tihao; Z. Yang Greedy Non-Intrusive Reduced Order Model for Fluid Dynamics, AIAA J., Volume 56 (2018), p. 12 | DOI

[10] X. Dai; Y. Maday 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] J. Dongarra; M. Gates; A. Haidar; J. Kurzak; P. Luszczek; S. Tomov; I. Yamazaki 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] M. Duarte; M. Massot; S. Descombes 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] A. Eghbal; A. G. Gerber; E. Aubanel Acceleration of unsteady hydrodynamic simulations using the parareal algorithm, J. Comput. Sci., Volume 19 (2017), pp. 57-76 | DOI

[14] C. Farhat; J. Cortial; C. Dastillung; H. Bavestrello 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] P. F. Fischer; F. Hecht; Y. Maday, Domain Decomposition Methods in Science and Engineering (2005), pp. 433-440 | DOI | Zbl

[16] M. J. Gander 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] M. J. Gander, Multiple Shooting and Time Domain Decomposition Methods (2015), pp. 69-113 | DOI | Zbl

[18] M. J. Gander; M. Petcu Analysis of a Krylov subspace enhanced parareal algorithm for linear problems, ESAIM, Proc., Volume 25 (2008), pp. 114-129 | DOI | MR | Zbl

[19] V. Guinot; S. Soares-Frazão 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] R. Haynes 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] M. Iizuka; K. Ono 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] J.-L. Lions; Y. Maday; G. Turinici 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] Yvon Maday 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] M. Müller On the POD method: an abstract investigation with applications to reduced-order modeling and suboptimal control, Ph. D. Thesis, Georg-August-Universität (2008)

[25] B. W. Ong; J. B. Schroder Applications of time parallelization, Comput. Vis. Sci., Volume 23 (2020) no. 1-4, 11 | DOI | MR

[26] G. Pagès; O. Pironneau; G. Sall The parareal algorithm for American options, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 11, pp. 1132-1138 | DOI | MR | Zbl

[27] C. W. Rowley 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] D. Ruprecht Wave propagation characteristics of Parareal, Comput. Vis. Sci., Volume 19 (2018), pp. 1-17 | DOI | MR | Zbl

[29] D. Ruprecht; R. Krause Explicit parallel-in-time integration of a linear acoustic-advection system, Computers Fluids, Volume 59 (2012), pp. 72-83 | DOI | MR | Zbl

[30] H. Samuel Time domain parallelization for computational geodynamics, Geochemistry, Geophysics, Geosystems, Volume 13 (2012) no. 1 | DOI

[31] R. Ştefănescu; I. M. Navon POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model, J. Comput. Phys., Volume 237 (2013), pp. 95-114 | DOI | MR | Zbl

[32] J. M. F. da Trindade; J. F. Pereira 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

Cited by Sources: