A Comprehensive Numerical Overview of the Performance of Godunov Solutions Using Roe and Rusanov Schemes Applied to Dam-Break Flow

Abstract

As open channel simulations are of great economic and human significance, many numerical approaches have been developed, with the Godunov schemes showing particular promise. To evaluate, confirm, and extend the simulation results of others, a variety of first- and second-order FVMs are available, with Rusanov and Roe schemes being used here to simulate the demanding case of 1D and 2D flows following a dam break. The virtual boundary cells approach is shown to achieve a monotonic solution for both interior and boundary cells, and while flux computation is employed at boundary cells, a refinement is only rarely used in existing models. A number of variations are explored, including the TVD MUSCL-Hancock (monotone upwind scheme for conservation laws) numerical scheme with several slope limiters in a quest to avoid spurious oscillations. The sensitivity of the results to both channel length and the ratio of downstream to initial upstream water depth is explored using 1D and 2D models. The Roe scheme with a Van Leer limiter as a slope limiter is shown to be both fast and slightly more accurate than other slope limiters for this problem, but the Rusanov scheme with different slope limiters works well for 1D simulations. Significantly, the selection of an appropriate slope limiter is shown to be best based on the ratio of the downstream to upstream water depth. However, this study focuses on the special case where the ratio of the initial depth downstream to upstream of the dam is equal to or less than 0.5, and these outcomes are compared to theoretical results. The 2D dam-break problem is used to further explore first- and second-order methods using different slope limiters, and the results show that the Superbee limiter can be problematic due to an observed large dispersion in depth contours. However, the most promising approaches from previous studies are confirmed to deserve the high regard given to them by many researchers.