Communications in Mathematical Sciences

Volume 18 (2020)

Number 8

Operator splitting based central-upwind schemes for shallow water equations with moving bottom topography

Pages: 2149 – 2168



Alina Chertock (Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

Alexander Kurganov (Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, China)

Tong Wu (Department of Mathematics, University of Texas, San Antonio, Tx., U.S.A.)


In this paper, we develop a robust and efficient numerical method for shallow water equations with moving bottom topography. The model consists of the Saint-Venant system governing the water flow coupled with the Exner equation for the sediment transport. One of the main difficulties in designing good numerical methods for such models is related to the fact that the speed of water surface gravity waves is typically much faster than the speed at which the changes in the bottom topography occur. This imposes a severe stability restriction on the size of time steps, which, in turn, leads to excessive numerical diffusion that affects the computed bottom structure. In order to overcome this difficulty, we develop an operator splitting approach for the underlying coupled system, which allows one to treat slow and fast waves in a different manner and using different time steps. Our method is based on the application of a finite-volume central-upwind scheme introduced in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5:133–160, 2007], and incorporates a staggered grid strategy needed for a proper approximation of the bottom topography function. A number of one- and two-dimensional numerical examples are presented to demonstrate the performance of the proposed method.


Saint Venant system of shallow water equations, moving bottom topography, Exner equation, operator splitting method, semi-discrete central-upwind schemes

2010 Mathematics Subject Classification

35L65, 65M08, 76M12, 86-08, 86A05

The work of A. Chertock was supported in part by NSF grant DMS-1818684. The work of A. Kurganov was supported in part by NSFC grant 11771201 and by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001).

Received 22 September 2019

Accepted 7 June 2020

Published 22 December 2020