We propose in the present work an iterative coupling method for a dimensionally heteregeneous problem. We consider the 3D linearized hydrostatic Navier–Stokes equations coupled with corresponding 2D linearized shallow water equations. We first show briefly how to derive the 2D linearized shallow water system from the 3D linearized hydrostatic Navier–Stokes system. Then we propose and study a Schwarz-like algorithm to couple the two systems and we prove under some assumptions that the convergence of this Schwarz algorithm is equivalent to the convergence of the classical domain decomposition algorithm of shallow water equations. Finally, we give some theoretical results related to the control of the difference between a global 3D reference solution and the 3D part of the coupled solution. These results are illustrated numerically.
Keywords: dimensionally heterogeneous coupling, domain decomposition, multiscale analysis, hydrostatic Navier–Stokes equations, shallow water equations
Manel Tayachi 1 ; Céline Acary-Robert 2 ; Éric Blayo 2
@article{SMAI-JCM_2026__12__1_0,
author = {Manel Tayachi and C\'eline Acary-Robert and \'Eric Blayo},
title = {Design and analysis of a {Schwarz} coupling method for {3D} {Navier{\textendash}Stokes} equations and {2D} {Shallow} {Water} equations},
journal = {The SMAI Journal of computational mathematics},
pages = {1--26},
year = {2026},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {12},
doi = {10.5802/jcm.141},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.141/}
}
TY - JOUR AU - Manel Tayachi AU - Céline Acary-Robert AU - Éric Blayo TI - Design and analysis of a Schwarz coupling method for 3D Navier–Stokes equations and 2D Shallow Water equations JO - The SMAI Journal of computational mathematics PY - 2026 SP - 1 EP - 26 VL - 12 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.141/ DO - 10.5802/jcm.141 LA - en ID - SMAI-JCM_2026__12__1_0 ER -
%0 Journal Article %A Manel Tayachi %A Céline Acary-Robert %A Éric Blayo %T Design and analysis of a Schwarz coupling method for 3D Navier–Stokes equations and 2D Shallow Water equations %J The SMAI Journal of computational mathematics %D 2026 %P 1-26 %V 12 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/jcm.141/ %R 10.5802/jcm.141 %G en %F SMAI-JCM_2026__12__1_0
Manel Tayachi; Céline Acary-Robert; Éric Blayo. Design and analysis of a Schwarz coupling method for 3D Navier–Stokes equations and 2D Shallow Water equations. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 1-26. doi: 10.5802/jcm.141
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