Communications in Mathematical Sciences

Volume 15 (2017)

Number 3

A large time-step and well-balanced Lagrange-projection type scheme for the shallow water equations

Pages: 765 – 788



Christophe Chalons (Laboratoire de Mathématiques de Versailles, Université Paris-Saclay, Versailles, France)

Pierre Kestener (Maison de la Simulation, Digiteo Labs, CEA Saclay, Gif-sur-Yvette, France)

Samuel Kokh (CEA Saclay, Gif-sur-Yvette, France)

Maxime Stauffert (Laboratoire de Mathématiques de Versailles, Université Paris-Saclay, Versailles, France)


This work focuses on the numerical approximation of the shallow water equations (SWE) using a Lagrange-projection type approach. We propose to extend to this context the recent implicit-explicit schemes developed in [C. Chalons, M. Girardin, and S. Kokh, SIAM J. Sci. Comput., 35(6):a2874–a2902, 2013], [C. Chalons, M. Girardin, and S. Kokh, Commun. Comput. Phys., to appear, 20(1):188–233, 2016] in the framework of compressible flows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the wellknown well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases.


Lagrange-projection schemes, implicit-explicit schemes, large time steps, well-balanced property

2010 Mathematics Subject Classification

35L40, 35Q35, 76M12

Published 24 February 2017