COMPOSITE NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS Part II. Local Combinations

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Abstract
In Part II local combinations of PDE solutions are considered, where the grid spacing is reduced in only the marching or evolving direction. Via a local combination of difference solutions, extended ranges of stability for the classical explicit (CE) and, analogously, Euler methods are found. The range of stability is extended (n + 1)2 times the original limit where “n” is the number of combinations. Accounting for the computing effort for this multistep composite, the computing effort can, at best, be cut in half by the composite over the original CE method. The TE of these methods is analyzed and local extrapolations are considered. Local composites of different analogs, such as the Saul'yev average (SA) method, are also considered. The composite of the Crank-Nicholson and CE methods is shown to be equivalent to the optimum implicit method but stable only for p < 1.5. A new composite of the SA and CE is presented and shown to be fourth order in Ax spacing and superior to the other methods considered at large grid spacings.

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