COMPOSITE NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS Part I. Global Extrapolation

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Abstract
Composite techniques are developed for numerical solutions of partial differential equations (PDE's) by the combinations of different solutions. The combination can be global or local depending on the usage of the composite. A global composite is one in which the composite solution is not utilized in any continued calculations of the different numerical solutions. Global composites are space by space combinations, i.e., the solutions are obtained over the entire space of independent variables. For local combinations, on the other hand, the composite is used in the continued calculations of the different solutions. Therefore, the local procedure is a line by line algorithm where the different solutions are combined and become the initial values for the continued calculations of the different solutions. Composites considered are (i) global extrapolation, (ii) local extrapolation and local stabilization, (iii) alternating direction methods, and (iv) acceleration. By extrapolation the solution over the space of independent variables is calculated and combined with solutions for successively smaller grid spacings, characterized by hm . Global extrapolation is applied to the finite difference solution of PDE's and shown to be effective for reducing the truncation error (TE) of fixed difference methods. On the basis of computation time the spacing sequence h0 (m + 1) is shown to reduce the TE more efficiently than fco/2m for more than two extrapolations, where /i0 characterizes the largest spacing.