Communications in Mathematical Sciences

Volume 19 (2021)

Number 5

Interaction of the elementary waves for shallow water equations with discontinuous topography

Pages: 1381 – 1402



Qinglong Zhang (School of Mathematics and Statistics, Ningbo University, Ningbo, China)

Wancheng Sheng (Department of Mathematics, Shanghai University, Shanghai, China)

Yuxi Zheng (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears when the water depth changes, especially when there exists a bottom step. In this paper, we are mainly concerned with the interaction between a stationary wave with either a shock wave or a rarefaction wave. By using the characteristic analysis methods, the evolution of waves is described during the interaction process. The solution in large time scale is also presented in each case. The results may contribute to research on more complicated wave interaction problems.


shallow water equations, source term, interaction of elementary waves, Riemann problem

2010 Mathematics Subject Classification

Primary 35L60, 35L65, 35L80, 35R35. Secondary 35L50.

This work is partially supported by NSFC 11771274.

Received 24 May 2020

Accepted 6 January 2021

Published 11 November 2021