Large Time Step TVD High Order Scheme for Shallow Water Equations

Abstract

The Large Time Step (LTS) scheme originally proposed by LeVeque in 1998 has led to the development of highly efficient computational solvers of the Shallow Water Equations (SWEs). However, the scheme can lead to spurious oscillations in the flow solution, which are difficult to cure. This paper examines the use of a Total Variation Diminishing (TVD) high order scheme in conjunction with LTS. We first apply the scheme to the solution of the homogeneous 1D SWEs and obtain satisfactory results for three cases, even though small oscillations nevertheless occur when the CFL number is very large. The TVD-LTS scheme is then implemented for the first time for well-balanced SWEs containing bed-slope source terms. We find that oscillations are not as suppressed as for the homogeneous SWEs when the TVD-LTS scheme is applied to 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 a CFL number 4 before the solution breaks down.

Keywords: shallow water equations, Large Time Step Scheme, TVD