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Quadratic Spline Galerkin Method for the Shallow Water Equations on the Sphere
, 2002
"... Currently in most global meteorological applications, the spectral transform method or loworder finite difference/finite element methods are used. The spectral transform method, which yields highorder approximations, requires Legendre transforms. The Legendre transforms have a computational comple ..."
Abstract

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Currently in most global meteorological applications, the spectral transform method or loworder finite difference/finite element methods are used. The spectral transform method, which yields highorder approximations, requires Legendre transforms. The Legendre transforms have a computational complexity of O(N ), 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 [2] with initial conditions chosen to generate a soliton.
Optimal Quadratic Spline Collocation Methods for the
 In preparation
, 2000
"... In this study, we present numerical methods, based on the optimal quadratic spline collocation (OQSC) methods, for solving the shallow water equations (SWEs) in spherical coordinates. A quadratic spline collocation method approximates the solution of a differential problem by a quadratic spline. In ..."
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In this study, we present numerical methods, based on the optimal quadratic spline collocation (OQSC) methods, for solving the shallow water equations (SWEs) in spherical coordinates. A quadratic spline collocation method approximates the solution of a differential problem by a quadratic spline. In the standard formulation, the quadratic spline is computed by making the residual of the differential equations zero at a set of collocation points; the resulting error is second order, while the error associated with quadratic spline interpolation is fourth order locally at certain points and third order globally. The OQSC methods generate approximations of the same order as quadratic spline interpolation. In the onestep OQSC method, the discrete differential operators are perturbed to eliminate loworder error terms, and a highorder approximation is computed using the perturbed operators. In the twostep OQSC method, a secondorder approximation is generated first, using the standard formulation, and then a highorder approximation is computed in a second phase by perturbing the right sides of the equations appropriately.