Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

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.

A novel technique for implementing the finite element method in a shallow water equation

“…In the derivation of the finite element scheme, we use a Galerkin procedure to approximate the weak form [ [11] , [26] ] by writing as a linear combination of conformal linear basis or standard linear basis and by writing as a linear combination of non-conformal linear basis or simply, discontinuous linear basis such that where denote nodal values and denote the number of segments and vertices of the triangles respectively. The discontinuous linear basis and standard linear basis is illustrated as in Fig.…”

“…The shallow water wave equation (SWE) is a partial differential equation that enables us to gain a deeper understanding of physical phenomena in shallow environments. Assuming the horizontal length scale is greater than the depth scale, the SWE widely use to describe various real-world phenomena, for example: standing waves [ [1] , [2] , [3] , [4] ], wave refraction [ 5 ], dam break [ 6 ], internal wave generation in the strait [ [7] , [8] , [9] , [10] ], tsunami propagation in near-shore areas [ 11 ], and etc.…”

A Momentum-Conserving Scheme for Flow Simulation in 1D Channel with Obstacle and Contraction

“…The same set up as in the previous is used, except that here we use a downstream water level h(L, t) = 0.40, and the narrowest width b m = b c that corresponds to the critical flow condition. In the case with the upstream Froude number F = 0.4, Equation (15) results in b c = 0.74. In this setting, the steady flow that is developed has the form of a transcritical flow, as shown in Figure 6b.…”

“…The formulation of this scheme is based on a discrete version of conservation of mass and momentum balance, from which it gains its name as the momentum conserving staggered-grid scheme, abbreviated as the MCS-scheme, see in [14] for detail. See also in [15] for the 2-dimensional version of the MCS-scheme. By evaluating the surface and velocity variables on adjacent staggered grid points, and applying the upwind approximation to calculate the appropriate flux, the scheme does not require any Riemann solver for calculating flux, so the numerical calculation is cheap.…”

The Momentum Conserving Scheme for Two-Layer Shallow Flows

“…The scheme can be used to solve problems with rapidly varying flows, such as those involving hydraulic jumps and bores. This scheme has been implemented and extended to various shallow water flow problems, such as in [10,16,17] where it was referred to as the MCS scheme, an abbreviation for the momentum conserving staggered-grid scheme. The extension of the MCS scheme to the two-layer model has been successfully used for the study of internal waves in [18].…”