Paul N. Swarztrauber

A standard test set for numerical approximations to the shallow water equations in spherical geometry

A suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry. The shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth. These cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade-offs which must always be made in numerical modeling. Before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods. The cases are presented in order of complexity. They consist of advection across the poles, steady state geostrophically balanced flow of both global and local scales, forced nonlinear advection of an isolated low, zonal flow impinging on an isolated mountain, Rossby-Haurwitz waves, and observed atmospheric states. One of the cases is also identified as a computer performance/algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers.

A direct Method for the Discrete Solution of Separable Elliptic Equations

This paper extends the direct method of cyclic reduction to linear systems which result from the discretization of separable elliptic equations with Dirichlet, Neumann, or periodic boundary conditions. For an

Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3]

m \times n

Vectorizing the FFTs

net, the operation count is proportional to

SPHEREPACK 3.0: A Model Development Facility

mn\log _2 n

Efficient FORTRAN subprograms for the solution of elliptic partial differential equations (abstract)

and

On the Spectral Approximation of Discrete Scalar and Vector Functions on the Sphere

mn

Efficient FORTRAN subprograms for the solution of elliptic partial differential equations.

storage locations are required.

The Direct Solution of the Discrete Poisson Equation on a Disk

Computer models of geophysical processes often require the numerical solution of elliptic partial differential equations. This is particularly true for models that make use of stream functions, velocity potentials, or vorticity equations and for models that compute the pressure of an incompressible fluid. The numerical solution of elliptic equations can be a formidable programming task. Moreover, the equations are often time dependent, requiring repeated solutions and, hence, considerable computing resources. Recent advances in computing methods [1, 2] made it possible to solve a very large class of elliptic equations (the separable ones) rapidly and with minimal storage. And as a result of work on singular problems [6, 8], this class is free of special cases for which solutions cannot be obtained numerically. This paper describes a package of computer programs that make use of current methods for solving elliptic partial differential equations. The package is fully documented in [7]. We fLrst became involved in implementing the Buneman algorithm [2] and its extensions via the capacitance matrix approach [1] for solving Poissons equation

Vector and parallel methods for the direct solution of Poisson's equation

Publisher Summary This chapter provides an overview on vectorizing the FFTs. The fast Fourier transform (FFT) is the most well known of all algorithms. It is superior to the slow transform and has applications in all areas of scientific computing. The term FFT was applied to a specific algorithm for the rapid computation of the discrete complex Fourier transform; however, it has become a generic term that is applied to any one of a large number of algorithms that compute the complex as well as other Fourier transforms. Many algorithms exist for a given Fourier transform, and when they are applied to a particular sequence, the result is the same. However, the algorithms differ in the ways that intermediate results are computed and stored. It is these important differences that provide the algorithms with unique properties that make one or the other more attractive for a particular application.