A 3-D implicit finite-volume model of shallow water flows

Abstract An icosahedral-hexagonal shallow-water model (SWM) on the sphere is formulated on a local Cartesian coordinate based on the general stereographic projection plane. It is discretized with the third-order Adam–Bashforth time-differencing scheme and the second-order finite-volume operators for spatial derivative terms. The finite-volume operators are applied to the model variables defined on the nonstaggered grid with the edge variables interpolated using polynomial interpolation. The projected local coordinate reduces the solution space from the three-dimensional, curved, spherical surface to the two-dimensional plane and thus reduces the number of complete sets of basis functions in the Vandermonde matrix, which is the essential component of the interpolation. The use of a local Cartesian coordinate also greatly simplifies the mathematic formulation of the finite-volume operators and leads to the finite-volume integration along straight lines on the plane, rather than along curved lines on the spherical surface. The SWM is evaluated with the standard test cases of Williamson et al. Numerical results show that the icosahedral SWM is free from Pole problems. The SWM is a second-order finite-volume model as shown by the truncation error convergence test. The lee-wave numerical solutions are compared and found to be very similar to the solutions shown in other SWMs. The SWM is stably integrated for several weeks without numerical dissipation using the wavenumber 4 Rossby–Haurwitz solution as an initial condition. It is also shown that the icosahedral SWM achieves mass conservation within round-off errors as one would expect from a finite-volume model.

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A finite volume method for steady state 2D shallow water flows

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/09615539710156174 ◽

1997 ◽

Vol 7 (1) ◽

pp. 4-23 ◽

Cited By ~ 6

Author(s):

J.G. Zhou ◽

I.M. Goodwill

Keyword(s):

Steady State ◽

Shallow Water ◽

Finite Volume Method ◽

Finite Volume ◽

Water Flows ◽

Shallow Water Flows ◽

Volume Method

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Quasi-two-layer finite-volume scheme for modelling shallow water flows over an arbitrary bed in the presence of external force. II. Algorithm applications and numerical results

Russian Journal of Numerical Analysis and Mathematical Modelling ◽

10.1515/rnam-2014-0014 ◽

2014 ◽

Vol 29 (3) ◽

Author(s):

Kirill V. Karelsky ◽

Arakel S. Petrosyan ◽

Alexander G. Slavin

Keyword(s):

Shallow Water ◽

Finite Volume ◽

External Force ◽

Large Scale ◽

Lower Layer ◽

Three Dimensional ◽

Finite Volume Scheme ◽

Water Flows ◽

Shallow Water Flows ◽

Scale Motion

AbstractA finite-volume numerical method for studying shallow water flows over an arbitrary bed profile in the presence of external force has been proposed in [33]. This method uses the quasi-two-layer model of hydrodynamic flows over a stepwise boundary with advanced consideration of the flow features near the step. A distinctive feature of the suggested model is a separation of the studied flow into two layers in calculating the flow quantities near each step, and improving by this means the approximation of depth-averaged solutions of the initial three-dimensional Euler equations. We are solving the shallow-water equations for one layer, introducing the fictitious lower layer only as an auxiliary structure in setting up the appropriate Riemann problems for the upper layer. Besides, the quasi-two-layer approach leads to the appearance of additional terms in the one-layer finite-difference representation of balance equations. Numerical simulations are performed based on the proposed in [33] algorithm of various physical phenomena, such as breakdown of the rectangular fluid column over an inclined plane, large-scale motion of fluid in the gravity field in the presence of Coriolis force over amounted obstacle on the underlying surface. Computations are made for the two-dimensional dam-break problem on a slope precisely conform to laboratory experiments. The interaction of the Tsunami wave with the shore line including an obstacle has been simulated to demonstrate the efficiency of the developed algorithm in domains, including partly flooded and dry regions.

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