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Vectorization of Diffusion Computations in the Presence of Periodic Boundary Conditions

I. K. Abu-Shumays

Nuclear Science and Engineering

Volume 105 / Number 1 / May 1990 / Pages 40-51


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Effective utilization of translational or rotational periodic boundary conditions, when applicable, can substantially reduce the cost of solving very large multidimensional elliptic diffusion problems. Application of periodic boundary conditions, however, perturbs the overall matrix structure of the underlying discretized diffusion equations, and special care should be exercised to avoid loss of computational efficiency. For simplicity, only the numerical solution of two-dimensional diffusion problems is discussed. Developing and testing on a vector computer alternative algorithms for implementing periodic boundary conditions within the framework of point and line iteration methods are described. For illustration, only the point Chebyshev and red-black line cyclic Chebyshev iterative methods are considered. Vectorization methods previously developed are extended to allow for periodic boundary conditions. The method of odd-even cyclic reduction as applied to vectorization of the solution of tridiagonal systems is generalized to apply to special matrix equations that are almost of tridiagonal form. Consequently, it is demonstrated numerically on a CYBER 205 computer for model two-dimensional problems that the resulting red-black line cyclic Chebyshev iterative method is computationally superior to the highly vectorizable point Chebyshev iterative method. The superiority of the red-black line methods over the point methods is expected to hold for more complex problems with general mesh triangulations.

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