Abstract
In this study, attention is focused on numerically solving the two-dimensional, two-layer nonlinear shallow-water equations (2LSWEs) over uneven bed topography. A two-layer hydrostatic-reconstruction method (2LHRM) is proposed for face value reconstructions. A numerical model is then developed based on the 2LHRM in the framework of finite-volume methods using the slope-limited centered (SLIC) approximate Riemann solver for flux evaluations. The validations against various benchmark tests show that the 2LHRM by working with the SLIC scheme predicts robust solutions around large gradients and is able to simulate lake-at-rest solutions without using any ad-hoc techniques.
Highlights
The last two decades have seen extensive use of upwind Godunov-type and centered finite-volume methods to solve the one-layer nonlinear shallow-water equations, of which the former is dominantly used to date
In solving the 1LSWEs, the hydrostatic-reconstruction method (HRM) proposed by Audusse et al [3] is widely adopted; in this method, singlevalued inter-cell bed elevations are dynamically calculated based on the reconstructed inter-cell water surface elevation and water depth values
As the HRM gains popularity and success in solving the 1LSWEs, while we found none of existing 2LSWEs solvers is developed based on HRM, we are curious that can HRM be extended for solving the 2LSWEs, and if so, how does it perform in terms of capturing discontinues and maintaining steady stationary flow at rest? Besides, lots of existing 2LSWEs solvers are developed based on upwind Godunov-type method which relies on evaluation of eigenvalues of the partial-differential equation system
Summary
The last two decades have seen extensive use of upwind Godunov-type and centered finite-volume methods to solve the one-layer nonlinear shallow-water equations (abbreviated as 1LSWEs hereafter), of which the former is dominantly used to date. Spinewine et al [10] proposed an ad-hoc technique to switch from solving the water-level based continuity equation under a quiescent flow condition (QFC), to solving the water-depth based continuity equation not under a QFC This method is able to numerically maintain quiescent flow at rest, yet it violates the principle of mass conservation because the temporal variation of the interface is unconsidered. Izem et al [13] developed a finite-element discontinuous Galerkin method for solving the 2LSWEs with the local Lax–Friedrichs scheme for flux evaluations; their numerical tests show that spurious oscillations may be generated near the discontinuities even for a flat bed situation. (cu)pper-layer surface lower-layer surface (interface) h2 lower-layer bottom (bed) z2 h1 z1 zb zr Figure 1: Definition sketch of a two-layer shallow-water system (not to scale)
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