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

TL;DR

This paper analyzes how stationary waves interact with shock and rarefaction waves in shallow water equations with discontinuous topography, providing insights into wave evolution and long-term behavior.

Contribution

It offers a detailed analysis of wave interactions involving stationary waves in shallow water equations with discontinuous bottom topography, expanding understanding of complex wave dynamics.

Findings

Wave interactions are characterized using characteristic analysis methods.

The evolution of waves during interactions is described in detail.

Long-term solutions for wave interactions are provided.

Abstract

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.