Parallel Implementation of a Data-Transpose Technique for the Solution of Poisson's Equation in Cylindrical Coordinates

We present a parallel nite-diierence algorithm for the solution of the 3D cylindrical Poisson equation. The algorithm is based on a data-transpose technique, in which all computations are performed independently on each node, and all communications are restricted to global 3D data-transposition between nodes. The data-transpose technique aids us in implementing two sequential algorithms for the solution of the cylindrical Poisson equation. The rst algorithm is based on the alternating direction implicit (ADI) method and the second is based on Fourier Analysis (FA). In both algorithms we rst perform a Fourier transform in the naturally periodic azimuthal coordinate direction. This decouples the 3D problem into a set of independent 2D problems which are then solved by the ADI and FA methods. In the ADI method we convert each 2D problem into two sets of 1D (tridiagonal) problems which are then solved iteratively by alternating in the radial and vertical coordinate directions. In the FA method we convert each 2D problem into a set of tridiagonal problems by transforming into an orthogonal \Fourier" basis on whose type depends on the boundary conditions in the vertical coordinate direction. In both algorithms, the resulting tridiagonal systems are solved using LU decomposition with forward-and back-substitution. An operation and communication count analysis of these algorithms plus an internal timing analysis of the code is presented. Execution times are measured and compared on a 4K-node massively parallel MasPar MP-2 and on a single processor Cray C90 vector machine. Experimental results show that the new algorithm is eecient and provides good performance on the inexpensive MasPar machine due to it's high communication to computation ratio.