Operator splitting based central-upwind schemes for shallow water equations with moving bottom topography
The authors study the shallow water system with time-dependent bottom topography. They consider the simplest case, in which the shallow water waves are modeled by the Saint-Venant system and the bottom topography movement is governed by the Exner equation. The resulting system is hyperbolic and its Jacobian matrices typically contain both large and small eigenvalues. The largest eigenvalues determine the speed of the fast surface waves and the smallest eigenvalue reflects the (slow) speed of the bottom topography propagation. A finite volume method is presented to approximate the underlying system using an explicit scheme. The time steps, which are inversely proportional to the maximum of the spectral radius of the Jacobians over the entire computational domain, would be too small. An operator splitting approach is developed which allows to deal with large time steps in the bottom evolution stages, while keeping the small CFL-controlled time steps in the hydrodynamics stages. The method presented here is based on the application of a finite-volume central-upwind scheme introduced in [\textit{A. Kurganov} and \textit{G. Petrova}, Commun. Math. Sci. 5, No. 1, 133--160 (2007; Zbl 1226.76008)]. Some numerical tests in one- and two-dimensions are presented to demonstrate the performance of the proposed method.
Saint Venant system of shallow water equations
scientific article; zbMATH DE number 7342285