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Dispersion and stability analyses of the linearized two-dimensional shallow water equations in boundary-fitted co-ordinates. (English) Zbl 1143.76518
Summary: Fourier analysis is used to study the phase and group speeds of a linearized, two-dimensional shallow water equations, in a non-orthogonal boundary-fitted co-ordinate system. The phase and group speeds for the spatially discretized equations, using the second-order scheme in an Arakawa C grid, are calculated for grids with varying degrees of non-orthogonality and compared with those obtained from the continuous case. The spatially discrete system is seen to be slightly dispersive, with the degree of dispersivity increasing with an decrease in the grid non-orthogonality angle or decrease in grid resolution and this is in agreement with the conclusions reached by S. Sankaranarayanan and M. S. Spaulding [J. Comput. Phys. 184, No. 1, 299–320 (2003; Zbl 1118.76327)]. The stability condition for the non-orthogonal case is satisfied even when the grid non-orthogonality angle, is as low as 30\(^{\circ}\) for the Crank Nicolson and three-time level schemes. A two-dimensional wave deformation analysis, based on complex propagation factor developed by Leendertse (Report RM-5294-PR, The Rand Corp., Santa Monica, CA, 1967), is used to estimate the amplitude and phase errors of the two-time level Crank-Nicolson scheme. There is no dissipation in the amplitude of the solution. However, the phase error is found to increase, as the grid angle decreases for a constant Courant number, and increases as Courant number increases.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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