Dispersion and stability analyses of the linearized two-dimensional shallow water equations in Cartesian coordinates |
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| Authors: | S Sankaranarayanan Malcolm L Spaulding |
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| Institution: | a Applied Science Associates, 70 Dean Knauss Drive, Narragansett, RI 02882, USA;b Ocean Engineering, University of Rhode Island, Narragansett, RI 02882, USA |
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| Abstract: | In the present study, a Fourier analysis is used to develop expressions for phase and group speeds for both continuous and discretized, linearized two-dimensional shallow water equations, in Cartesian coordinates. The phase and group speeds of the discrete equations, discretized using a three-point scheme of second order, five-point scheme of fourth order and a three-point compact scheme of fourth order in an Arakawa C grid, are calculated and compared with the corresponding values obtained for the continuous system. The three-point second-order scheme is found to be non-dispersive with grid resolutions greater than 30 grids per wavelength, while both the fourth-order schemes are non-dispersive with grid resolutions greater than six grids per wavelength. A von Neumann stability analysis of the two- and three-time-level temporal schemes showed that both schemes are stable. A wave deformation analysis of the two-time-level Crank–Nicolson scheme for one-dimensional and two-dimensional systems of shallow water equations shows that the scheme is non- dispersive, independent of the Courant number and grid resolution used. The phase error or the dispersion of the scheme decreases with a decrease in the time step or an increase in grid resolution. |
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| Keywords: | Stability Dispersion Cartesian coordinates Compact difference |
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