A New Time Stepping Method for the Solution of the Shallow Water Equations on a Rotating Sphere

Ali Rouhi Institute for Nonlinear Science, University of California, San Diego, La Jolla, California

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Roy Schult Institute for Nonlinear Science, University of California, San Diego, La Jolla, California

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Jon Wright Institute for Nonlinear Science, University of California, San Diego, La Jolla, California

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Abstract

The shallow water equations on a rotating sphere form a system of great interest in geophysical fluid dynamics, from meteorology to atmospheric physics and climate studies. Williamson and his collaborators at the National Center for Atmospheric Research introduced a set of numerical experiments of increasing complexity to serve as a standard set for testing proposed numerical schemes. They also provided a software package that solved the equations by methods that are currently state of the art in the field. The authors recently introduced a new method (called “odd–even splitting”) for solving a wide variety of partial differential equations which has several advantages. In this paper a variation of this method, called the “dual-copy splitting” method of time stepping, is applied to several of the test cases of Williamson et al. using for the spatial discretization the same spectral transform model as they used. The method has the following features that are novel, so far as the authors know. First, the order of accuracy in time can be increased from the usual first or second order, without additional memory requirements to store fields at intermediate time steps. Second, gravity waves, which impose a prohibitively small time step for usual explicit schemes, have been handled here “exactly” while retaining the advantages of implicit schemes. Indeed, it is believed that this treatment of gravity waves is in fact superior to implicit methods.

Corresponding author address: Dr. Jon Wright, Institute for Nonlinear Science, University of California, San Diego, Mail Code 0402, La Jolla, CA 92093-0402.

Email: jawright@ucsd.edu

Abstract

The shallow water equations on a rotating sphere form a system of great interest in geophysical fluid dynamics, from meteorology to atmospheric physics and climate studies. Williamson and his collaborators at the National Center for Atmospheric Research introduced a set of numerical experiments of increasing complexity to serve as a standard set for testing proposed numerical schemes. They also provided a software package that solved the equations by methods that are currently state of the art in the field. The authors recently introduced a new method (called “odd–even splitting”) for solving a wide variety of partial differential equations which has several advantages. In this paper a variation of this method, called the “dual-copy splitting” method of time stepping, is applied to several of the test cases of Williamson et al. using for the spatial discretization the same spectral transform model as they used. The method has the following features that are novel, so far as the authors know. First, the order of accuracy in time can be increased from the usual first or second order, without additional memory requirements to store fields at intermediate time steps. Second, gravity waves, which impose a prohibitively small time step for usual explicit schemes, have been handled here “exactly” while retaining the advantages of implicit schemes. Indeed, it is believed that this treatment of gravity waves is in fact superior to implicit methods.

Corresponding author address: Dr. Jon Wright, Institute for Nonlinear Science, University of California, San Diego, Mail Code 0402, La Jolla, CA 92093-0402.

Email: jawright@ucsd.edu

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