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Dispersion and stability analyses of the linearized two-dimensional shallow water equations in Cartesian coordinates

机译：笛卡尔坐标系中线性化二维浅水方程的色散和稳定性分析

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摘要

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.
机译：在本研究中，使用傅立叶分析来为笛卡尔坐标系中连续和离散化的线性化二维浅水方程组开发相位和组速度的表达式。计算在Arakawa C网格中使用二阶三点方案，四阶五点方案和四阶三点紧凑方案离散的离散方程的相速度和群速度，并将其与连续系统获得的相应值。发现三点二阶方案是非色散的，其栅格分辨率大于每个波长30个栅格，而两个四阶方案都是非色散的，其栅格分辨率大于每个波长六个栅格。对两级和三级时间方案的冯·诺依曼稳定性分析表明，这两种方案都是稳定的。对一维和二维浅水方程组的二维Crank-Nicolson方案的波浪变形分析表明，该方案是非分散的，与所使用的Courant数和网格分辨率无关。相位误差或方案的离散度随时间步长的减小或网格分辨率的提高而减小。

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