A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

Abstract

We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant regime; multiple solutions in the resonant regime. This analysis leads us to a numerical algorithm that provides one with a Riemann solver. Next, we introduce a Godunov-type scheme based on this Riemann solver, which is well-balanced and of quasi-conservative form. Finally, we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime, except in the limiting situation when Riemann data precisely belong to the resonance hypersurface.

Central-upwind scheme for shallow water equations with discontinuous bottom topography

“…To demonstrate this, we compute the numerical solution until the final time t = 0.03 on two uniform grids with x = 0.004 and 0.001 and compare the obtained solution with a reference one computed with x = 0.00004. We plot the computed water depth h and velocity u in Figure 3.2, where one can clearly observe the convergence towards the reference solution, which agrees very well with the exact one (see [14,Test 7]). As it was shown in [14], the IVP (1.1), (3.2) admits three distinct analytic solution and Godunov-type upwind schemes based on different Riemann problem solvers converge to different analytic solutions.…”

“…More details can be seen in Example 2 -Riemann Problem with Unique Solution. In this example, we consider the Riemann problem from [14,Test 7], where the system (1.1) is solved with g = 9.8 and the following Riemann data:…”

“…In [14], the exact solution of the initial value problem (IVP) (1.1), (3.1) was obtained and it was also shown that the Godunov-type scheme based on the exact solution of the Riemann problem fails at this test. The central-upwind scheme proposed in this paper is, on the other hand, capable of accurately capturing the exact solution of the IVP (1.1), (3.1).…”

Appropriate model use for predicting elevations and inundation extent for extreme flood events

“…There are two major groups of shock-capturing schemes in use in computational fluid dynamics. One group of schemes is based on the Godunov method, which solves the Riemann problems at the interfaces of grid cells (Kerger et al 2011;LeFloch and Thanh 2011;Urbano and Nasuti 2013), while the other group is based on an arithmetical combination method of the first-and second-order upwind schemes Tseng 2004;Ouyang et al 2013). …”

A Study on Imposing Exact Solutions as Internal Boundary Conditions in Simulating Shallow-water Flows over a Step

“…단일 계단에 대한 정확해를 구할 때, 질량 보존과 함께 에너지의 보존을 고려하는 방법 (Alcrudo and Benkhaldoun, 2001;Chinnayya et al, 2004;LeFloch and Thanh, 2011)과 그 대신 운동량의 보존을 적용하는 방법 (Bernetti et al, 2008;Rosatti and Begnudelli, 2010 (internal boundary)로 처리한다 (Cunge et al, 1980;Jun, 1996;Jin and Fread, 1997). 그마저 어려운 경우에는 구조물을 연속 지형으로 간주하여 경사로 완화하게 되는데, 조석에 의한 역류가 발생되는 한강 하류의 신곡수중보가 그 사례이다 (Jun et al, 2007).…”