Keywords: shallow water, sphere, higher order finite differences, pseudospectral.

Shallow water equations (SWE) are widely used to represent phenomena in several fields of fluid dynamics. The SWE, which describe the flow of a thin layer of fluid in two dimensions have been used by the atmospheric modeling community as a vehicle for testing promising numerical methods for solving atmospheric and oceanic problems. The numerical solution of the SWE of a global atmospheric model provides a challenging computational problem. In this study we present a p-adaptive method for the solution of the shallow water on a sphere. Based on a set of standardized test cases the resulting model performance is investigated.

The shallow water equations (SWE), describing a thin layer of fluid flow (liquid or gas) in two dimensions, is the simplest possible model that captures the essential characteristics of fluid flow on a sphere.

There is a relatively large group of numerical methods for solving the SWE on a sphere, each of them having advantages and disadvantages concerning accuracy, efficiency, reliability, conservation, flexibility, scalability, etc. The numerous methods can be classified by the different spatial discretization.

One of the most popular class of methods for solving flow problems on a sphere is the spectral transform method (STM). The STM expresses the variables as series expansions in terms of the spherical harmonic basis functions, and is known for its high level of accuracy and for being extremely stable. Fornberg [1] completed a successful numerical analysis of a simpler set of fluid flow equations on a spherical geometry using a pseudospectral method.

The finite element method (FEM) is another commonly used method. Other commonly employed method in climate modeling, is the the finite volume method (FVM). By its very design criterion, the finite volume methods (FVM) guarantee conservation.

Numerical methods for solving partial differential equations use adaptively refined grids to increase the order of accuracy locally, thereby distributing the discretization error uniformly. Using the h-adaptive method in the case of solving various coupled partial differential equations, it is not possible to refine by employing the same grid for all the variables involved. Another adaptive method which can be employed using the same grid for all the variables, it is the p-adaptive method. In this method the approximation can be different for each one of the variables.

In this paper we present a p-adaptive method based in higher order finite differences and pseudospectral derivatives that is applied in the solution of the shallow water equations on a sphere. The numerical experiments confirm the expected high-order accuracy.

In this paper we also describe a formulation of the shallow water equations in rectangular coordinates. We describe the numerical methods used to represent a sphere and to solve the SWE. A standard test suite of several problems for evaluating numerical methods for the SWE in spherical geometry was proposed by Williamson et al. [2] in 1992, and accepted by the modeling community in order to compare newly proposed methods. We describe the first two test cases of Williamson and give the results. The first case in the suite tests the advective component of the numerical method in isolation. Test case 2 (which involves the full set of shallow water equations) has a steady state solution to the non-linear SWE. For both cases an error analysis is done comparing with Williamson's results.

1

B. Fornberg, D. Merrill, "Comparison of finite difference and pseudospectral methods for convective flow over a sphere", SIAM Geophysic. Res. Lett., 24, No 24, 3245-3248, 1997. doi:10.1029/97GL03272

2

D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, and P.N. Swarztrauber. "A Standard Test Set for Numerical Approximations to the Shallow Water Equations in Spherical Geometry", Journal of Computational Physics, 102(1):211-224, 1991. doi:10.1016/S0021-9991(05)80016-6

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