Abstract
Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating wave propagation and run-up in shallow areas. We implemented numerically the 2-dimensional Shallow Water Equations (SWE) on a staggered grid by applying the momentum conserving approximation in the advection terms. The numerical model is named MCS-2d. For simulations of wet–dry phenomena and wave run-up, a method called thin layer is used, which is essentially a calculation of the momentum deactivated in dry areas, i.e., locations where the water thickness is less than the specified threshold value. Efficiency and robustness of the scheme are demonstrated by simulations of various benchmark shallow flow tests, including those with complex bathymetry and wave run-up. The accuracy of the scheme in the calculation of the moving shoreline was validated using the analytical solutions of Thacker 1981, N-wave by Carrier et al., 2003, and solitary wave in a sloping bay by Zelt 1986. Laboratory benchmarking was performed by simulation of a solitary wave run-up on a conical island, as well as a simulation of the Monai Valley case. Here, the embedded-influxing method is used to generate an appropriate wave influx for these simulations. Simulation results were compared favorably to the analytical and experimental data. Good agreement was reached with regard to wave signals and the calculation of moving shoreline. These observations suggest that the MCS method is appropriate for simulations of varying shallow water flow.
Highlights
The shallow water equations (SWE) are applicable to a wide range of practical problems, ranging from ocean dynamics to flows due to the collapse of hydro dams
Extensive research on numerical models for SWE has been developed for a long time, and can generally be classified into three types, i.e., the finite element method [1], the finite difference method [2], or the finite volume method [3]
We start with a discussion and derivation of the numerical model called the momentum conserving staggered grid (MCS-2d) scheme, as well as the embedded wave influx algorithm which is important for the construction of transparent wave influx
Summary
The shallow water equations (SWE) are applicable to a wide range of practical problems, ranging from ocean dynamics to flows due to the collapse of hydro dams. Discretization of the nonlinear terms are based on a conservative principle; i.e., momentum conserving approximation for the advection terms and upwind for the non-linear term in mass conservation In this way, we do not need to apply the Riemann solver in the flux calculation, and the numerical computation of MCS-2d is relatively cheap. We show the capability of the MCS-2d scheme to simulate various benchmark tests, as well as its accuracy in computing the moving shoreline. We start with a discussion and derivation of the numerical model called the momentum conserving staggered grid (MCS-2d) scheme, as well as the embedded wave influx algorithm which is important for the construction of transparent wave influx. By conducting validation with several benchmark tests, we demonstrate the robustness and capability of the MCS-2d scheme in predicting accurate results for various tsunami-related simulations
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