Communications in Mathematical Sciences

Volume 20 (2022)

Number 7

A second-order well-balanced Lagrange-projection numerical scheme for shallow water Exner equations in 1D and 2D

Pages: 1839 – 1873



Christophe Chalons (Université Paris-Saclay, Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), CNRS, Laboratoire de Mathématiques de Versailles, France)

Alessia del Grosso (Université Paris-Saclay, UVSQ, CNRS, Laboratoire de Mathématiques de Versailles, France)


The present work is devoted to the numerical approximation of the shallow water Exner system in both one and two dimensions, where the Exner equation expresses the evolution in time of the bed sediment. Both the Grass and the Meyer–Peter & Müller formulas are taken into account to model the solid transport discharge contributions. The numerical scheme is based on the Lagrange-projection formalism which consists in splitting the mathematical model into the acoustic and transport systems. This work is considered as a first step to understand how to include the Exner equation in this framework and, for this reason, the Exner equation is taken into account only at the transport level; both a decoupled and weakly coupled formulations are proposed. New strategies to include the Exner equation at the acoustic level or in both steps will be treated in the next work. The methods are designed in such a way to satisfy the well-balanced property as well. Details to reach the second-order of accuracy are given; numerical results are shown to validate the numerical schemes.


Lagrange-projection decomposition, shallow water equations, Exner equation, nonconservative hyperbolic systems, second order of accuracy, well-balanced property

2010 Mathematics Subject Classification

35L60, 65N08, 76M12

Received 12 March 2021

Received revised 11 January 2022

Accepted 27 January 2022

Published 21 October 2022