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Manuscript Title: TDPOIS, a vector-processor routine for the solution of the three- dimensional Poisson and biharmonic equations in a rectangular prism.
Authors: G.A. Houseman
Program title: TDPOIS
Catalogue identifier: AATC_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 43(1987)257
Programming language: Fortran.
Computer: CDC CYBER 205.
Operating system: VSOS.
RAM: 190K words
Word size: 64
Keywords: Fluid dynamics, Poisson, Vector processor, Biharmonic, Three dimensional, Fourier transform, Cyclic reduction, Potential theory.
Classification: 12.

Nature of problem:
The routine TDPOIS solves Poisson's equation or the biharmonic equation a rectangular prism. Poisson's equation occurs in many branches of potential theory but TDPOIS was developed for use in solving the equations of convective flow in a rectangular prism.

Solution method:
The source function is expanded in a Fourier sine or cosine series in each of the x and y-directions. The solution for each harmonic component is then obtained by cyclic reduction in the z-direction followed by Fourier synthesis in the x and y-directions. The Fourier transform and cyclic reduction algorithms are based on scalar algorithms developed for solution of the 2-dimensional Poisson equation, Comp. Phys. Commun 2(1971)127 and 2(1971)139, here modified to make optimum use of the vector processor of the Cyber 205.

The three-dimensional array on which the potential function is represented has dimensions (NX1, NY1, NZ1), where each of NX1, NY1 and NZ1 is the form N=2**m +1. The three dimensions are not necessarily equal, but the sampling interval is the same in each direction. TDPOIS only permits zero value boundary conditions in the z direction and either zero-value or zero-gradient boundary conditions in each of the x and y directions (independently). These restrictions could be relaxed to some degree by the use of more general Fourier transform and/or cyclic reduction algorithms than are employed here.

Unusual features:
The algorithm consists of a sequence of Fourier transform and cyclic reduction steps separated by matrix transpositions. Each transposition ensures that the next step is able to use the longest practicable vectors, for optimum efficiency of the Cyber 205 vector processor. These vectors are planes of the prism which are orthogonal to the direction of the operation performed in that step, and the term 'orthogonal vectorisation' is used to refer to this choice of element ordering in the matrix. The program is written in Cyber 200 FORTRAN language, allowing the extensive use of vector syntax and reference to vector intrinsic functions.

Running time:
Execution time for TDPOIS is approximately 31.1 ms for the Poisson equation and 34.2 ms for the biharmonic equation. These are the averages for 2000 calls to TDPOIS for a 33*33*33 matrix on a CDC Cyber 205 (2 pipeline) with 64 bit precision.