Math @ Duke

Publications [#243639] of Anita T. Layton
Papers Published
 Layton, AT; Christara, CC; Jackson, KR, Quadratic spline methods for the shallow water equations on the sphere: Galerkin,
Mathematics and Computers in Simulation, vol. 71 no. 3
(2006),
pp. 175186, ISSN 03784754 [doi]
(last updated on 2017/12/14)
Abstract: Currently in most global meteorological applications, loworder finite difference or finite element methods, or the spectral transform method are used. The spectral transform method, which yields highorder approximations, requires Legendre transforms. The Legendre transforms have a computational complexity of O ( N3 ), where N is the number of subintervals in one dimension, and thus render the spectral transform method unscalable. In this study, we present an alternative numerical method for solving the shallow water equations (SWEs) on a sphere in spherical coordinates. In this implementation, the SWEs are discretized in time using the twolevel semiLagrangian semiimplicit method, and in space on staggered grids using the quadratic spline Galerkin method. We show that, when applied to a simplified version of the SWEs, the method yields a neutrally stable solution for the meteorologically significant Rossby waves. Moreover, we demonstrate that the Helmholtz equation arising from the discretization and solution of the SWEs should be derived algebraically rather than analytically, in order for the method to be stable with respect to the Rossby waves. These results are verified numerically using Boyd's equatorial wave equations [J.P. Boyd, Equatorial solitary waves. Part I. Rossby solitons, J. Phys. Oceanogr. 10 (1980) 16991717] with initial conditions chosen to generate a soliton. © 2006.


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