Application of Large Time Step TVD High Order Scheme to Shallow Water Equations

Abstract

The numerical modeling of actual river floods faces three challenges related to computational efficiency, accuracy, and proper balancing of terms in the governing equations, all of which are discussed in this paper. Herein, a large time step (LTS) scheme is used to improve efficiency, a high order scheme is used to enhance accuracy, and specific treatment of the bed slope term achieves a well-balanced form of shallow water equations. The LTS scheme, originally proposed by LeVeque in 1998, has led to the development of highly efficient computational solvers of the shallow water equations (SWEs). This paper examines use of a total variation diminishing (TVD) high order scheme in conjunction with LTS. We first applied the scheme to the solution of the homogeneous 1D SWEs and obtained satisfactory results for three cases, even though small oscillations nevertheless occur when the CFL number is very large. The additional source term makes the issue more complicated and can introduce a spurious flow when the term is not correctly handled. Many methods have been developed in traditional differential schemes, but not all are fit for the TVD-LTS scheme; for example, the method of decomposing the source term into simple characteristic waves has proved infeasible. In this paper the TVD-LTS scheme was implemented for the first time for well-balanced SWEs containing bed slope source terms. We found that oscillations were not as suppressed as for the homogeneous SWEs when the TVD-LTS scheme was applied to the three cases of step Riemann problems (SRP) tested for CFL numbers 1 to 10. For free surface flow over a bed hump, the TVD-LTS scheme can only reach CFL number 4 before the solution breaks down.