Spectral Transform Methods for Solving the Shallow-Water Equations on the Sphere

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Abstract
The accuracy of computed solutions to several formulations of the shallow-water equations is compared. The shallow-water equations can be written in a number of different forms that are obtained by (a) combining terms into differential expressions; (b) raising the order of the differential equations, for example, the vorticity–divergence formulation, and (c) transforming both independent and dependent variables. Although the exact solutions to the formulations are identical, the properties of the computed solutions vary depending on the formulation and the method of solution. Nine methods are examined, all of which provide satisfactory accuracy for a steady-state test case. The exact solution corresponds to a steady zonal wind that is tilted relative to the computational spherical coordinate system. The resulting wind passes over the North Pole, providing a test of the “pole problem.” Computational details are presented as well as the accuracy of a 5-day computation in both 32- and 64-bit arithme... Abstract The accuracy of computed solutions to several formulations of the shallow-water equations is compared. The shallow-water equations can be written in a number of different forms that are obtained by (a) combining terms into differential expressions; (b) raising the order of the differential equations, for example, the vorticity–divergence formulation, and (c) transforming both independent and dependent variables. Although the exact solutions to the formulations are identical, the properties of the computed solutions vary depending on the formulation and the method of solution. Nine methods are examined, all of which provide satisfactory accuracy for a steady-state test case. The exact solution corresponds to a steady zonal wind that is tilted relative to the computational spherical coordinate system. The resulting wind passes over the North Pole, providing a test of the “pole problem.” Computational details are presented as well as the accuracy of a 5-day computation in both 32- and 64-bit arithme...

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