Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients
Clement, Simon1 ; Lemarié, Florian1 ; Blayo, Eric1
1 Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France
The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 99-124.

In this paper, we investigate the effect of the space and time discretisation on the convergence properties of Schwarz Waveform Relaxation (SWR) algorithms. We consider a reaction-diffusion problem with discontinuous coefficients discretised on two non-overlapping domains with several numerical schemes (in space and time). A methodology to determine the rate of convergence of the classical SWR method with standard interface conditions (Dirichlet-Neumann or Robin-Robin) accounting for discretisation errors is presented. We discuss how such convergence rates differ from the ones derived at a continuous level (i.e. assuming an exact discrete representation of the continuous problem). In this work we consider a second-order finite difference scheme and a finite volume scheme based on quadratic spline reconstruction in space, combined with either a simple backward Euler scheme or a two-step “Padé” scheme (resembling a Diagonally Implicit Runge Kutta scheme) in time. We prove those combinations of space-time schemes to be unconditionally stable on bounded domains. We illustrate the relevance of our analysis with specifically designed numerical experiments.

Publié le :
DOI : 10.5802/smai-jcm.81
Classification : 65B99, 65L12, 65M12
Mots clés : Schwarz methods, Waveform relaxation, Semi-discrete
Clement, Simon 1 ; Lemarié, Florian 1 ; Blayo, Eric 1

1 Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France 
@article{SMAI-JCM_2022__8__99_0,
     author = {Clement, Simon and Lemari\'e, Florian and Blayo, Eric},
     title = {Discrete analysis of {Schwarz} waveform relaxation for a diffusion reaction problem with discontinuous coefficients},
     journal = {The SMAI Journal of computational mathematics},
     pages = {99--124},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {8},
     year = {2022},
     doi = {10.5802/smai-jcm.81},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/smai-jcm.81/}
}
TY  - JOUR
AU  - Clement, Simon
AU  - Lemarié, Florian
AU  - Blayo, Eric
TI  - Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients
JO  - The SMAI Journal of computational mathematics
PY  - 2022
SP  - 99
EP  - 124
VL  - 8
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - http://www.numdam.org/articles/10.5802/smai-jcm.81/
DO  - 10.5802/smai-jcm.81
LA  - en
ID  - SMAI-JCM_2022__8__99_0
ER  - 
%0 Journal Article
%A Clement, Simon
%A Lemarié, Florian
%A Blayo, Eric
%T Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients
%J The SMAI Journal of computational mathematics
%D 2022
%P 99-124
%V 8
%I Société de Mathématiques Appliquées et Industrielles
%U http://www.numdam.org/articles/10.5802/smai-jcm.81/
%R 10.5802/smai-jcm.81
%G en
%F SMAI-JCM_2022__8__99_0
Clement, Simon; Lemarié, Florian; Blayo, Eric. Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients. The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 99-124. doi : 10.5802/smai-jcm.81. http://www.numdam.org/articles/10.5802/smai-jcm.81/

[1] Azimzadeh, P.; Forsyth, P. A. Weakly Chained Matrices, Policy Iteration, and Impulse Control, SIAM J. Numer. Anal., Volume 54 (2016) no. 3, pp. 1341-1364 | DOI | MR | Zbl

[2] Al-Khaleel, M. D.; Wu, S.-L. Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem, Comput. Methods Appl. Math., Volume 20 (2020) no. 3, pp. 397-417 | DOI | MR | Zbl

[3] Alexander, R. Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s, SIAM J. Numer. Anal., Volume 14 (1977) no. 6, pp. 1006-1021 | DOI | MR | Zbl

[4] Britton, N. F. et al. Reaction-diffusion equations and their applications to biology., Academic Press Inc., 1986 | MR

[5] Berthe, P.-M. Méthodes de décomposition de domaine de type relaxation d’ondes optimisées pour l’équation de convection-diffusion instationnaire discrétisée par volumes finis, Ph. D. Thesis, Paris 13 (2013) http://www.theses.fr/2013pa132055 (Thèse de doctorat dirigée par Omnes, P. et Japhet, C. Mathématiques appliquées Paris 13 2013)

[6] Berthe, P.-M.; Japhet, C.; Omnes, P. Space–Time Domain Decomposition with Finite Volumes for Porous Media Applications, Domain Decomposition Methods in Science and Engineering XXI, Springer (2014), pp. 567-575 | DOI | Zbl

[7] Beerends, R. J.; ter Morsche, H. G.; van den Berg, J. C.; van de Vrie, E. M. Fourier and Laplace Transforms, Cambridge University Press, 2003 | DOI

[8] Caetano, F.; Gander, M. J.; Halpern, L.; Szeftel, J. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations, Netw. Heterog. Media, Volume 5 (2010) no. 3, pp. 487-505 | DOI | MR | Zbl

[9] Clement, Simon Code for Discrete analysis of SWR for a diffusion reaction problem with discontinuous coefficients, 2022 (https://zenodo.org/record/6324930) | DOI

[10] Gander, M. J. A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., Volume 6 (1999) no. 2, pp. 125-145 | DOI | MR | Zbl

[11] Gander, M. J.; Halpern, L. Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems, SIAM J. Numer. Anal., Volume 45 (2007) no. 2, pp. 666-697 | DOI | MR | Zbl

[12] Gander, M. J.; Halpern, L.; Hubert, F.; Krell, S. Optimized Overlapping DDFV Schwarz Algorithms, Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, Springer (2020), pp. 365-373 | DOI | Zbl

[13] Gander, M. J.; Halpern, L.; Kern, M. A Schwarz Waveform Relaxation Method for Advection—Diffusion—Reaction Problems with Discontinuous Coefficients and Non-matching Grids, Domain Decomposition Methods in Science and Engineering XVI, Springer (2007), pp. 283-290 | DOI

[14] Gander, M. J.; Hubert, F.; Krell, S. Optimized Schwarz Algorithms in the Framework of DDFV Schemes, Domain Decomposition Methods in Science and Engineering XXI, Springer (2014), pp. 457-466 | DOI | Zbl

[15] Gander, M. J.; Halpern, L.; Nataf, F. Optimal Schwarz waveform relaxation for the one dimensional wave equation, SIAM J. Numer. Anal., Volume 41 (2003) no. 5, pp. 1643-1681 | DOI | MR | Zbl

[16] Gander, M. J.; Kwok, F.; Mandal, B. C. Dirichlet–Neumann waveform relaxation methods for parabolic and hyperbolic problems in multiple subdomains, BIT Numer. Math., Volume 61 (2021) no. 1, pp. 173-207 | DOI | MR | Zbl

[17] Haynes, R. D.; Mohammad, K. Fully Discrete Schwarz Waveform Relaxation on Two Bounded Overlapping Subdomains, Domain Decomposition Methods in Science and Engineering XXV, Springer (2020), pp. 159-166 | DOI

[18] Kobayashi, M. H. On a Class of Padé Finite Volume Methods, J. Comput. Phys., Volume 156 (1999) no. 1, pp. 137-180 | DOI | MR | Zbl

[19] Lemarié, F.; Debreu, L.; Madec, G.; Demange, J.; Molines, J. M.; Honnorat, M. Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations, Ocean Modelling, Volume 92 (2015), pp. 124-148 | DOI

[20] Lemarié, F. Algorithmes de Schwarz et couplage océan-atmosphère, Theses, Université Joseph-Fourier - Grenoble I (2008) https://tel.archives-ouvertes.fr/tel-00343501

[21] Monge, A.; Birken, P. A Multirate Neumann–Neumann Waveform Relaxation Method for Heterogeneous Coupled Heat Equations, SIAM J. Sci. Comput., Volume 41 (2019) no. 5, p. S86-S105 | DOI | MR | Zbl

[22] Manfredi, G.; Ottaviani, M. Finite-difference schemes for the diffusion equation, Dynamical Systems, Plasmas and Gravitation, Springer (1999), pp. 82-92 | Zbl

[23] Nataf, F. Recent Developments on Optimized Schwarz Methods, Domain Decomposition Methods in Science and Engineering XVI, Springer (2007), pp. 115-125 | DOI | MR

[24] Nourtier-Mazauric, E.; Blayo, E. Towards efficient interface conditions for a Schwarz domain decomposition algorithm for an advection equation with biharmonic diffusion, Appl. Numer. Math., Volume 60 (2010) no. 1, pp. 83-93 https://www.sciencedirect.com/science/article/pii/s0168927409001652 | DOI | MR | Zbl

[25] Shchepetkin, A. F. An adaptive, Courant-number-dependent implicit scheme for vertical advection in oceanic modeling, Ocean Modelling, Volume 91 (2015), pp. 38-69 https://www.sciencedirect.com/science/article/pii/s1463500315000530 | DOI

[26] Smoller, J. Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften, 258, Springer, 1983 | DOI | Zbl

[27] Thery, S.; Pelletier, C.; Lemarié, F.; Blayo, E. Analysis of Schwarz waveform relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients, Numer. Algorithms (2021) | DOI

[28] Wu, S.-L.; Al-Khaleel, M. D. Semi-discrete Schwarz waveform relaxation algorithms for reaction diffusion equations, BIT Numer. Math., Volume 54 (2014) no. 3, pp. 831-866 | MR | Zbl

[29] Wu, S.-L.; Al-Khaleel, M. D. Optimized waveform relaxation methods for RC circuits: discrete case, ESAIM: M2AN, Volume 51 (2017) no. 1, pp. 209-223 | DOI | MR | Zbl

[30] Wood, N.; Diamantakis, M.; Staniforth, A. A monotonically-damping second-order-accurate unconditionally-stable numerical scheme for diffusion, Quarterly Journal of the Royal Meteorological Society, Volume 133 (2007) no. 627, pp. 1559-1573 | arXiv | DOI

[31] Zisowsky, A.; Ehrhardt, M. Discrete transparent boundary conditions for parabolic systems, Math. Comput. Modelling, Volume 43 (2006) no. 3, pp. 294-309 | DOI | MR | Zbl

Cité par Sources :