Optimal use of a numerical method for solving differential equations based on Taylor series expansions

Abstract

SUMMARYEfficiency in solving differential equations is improved by increasing the order of a Taylor series approximation. Computing time can be reduced up to a factor of 40 and an amount of memory storage can be saved, up to a factor of 70.The truncation error can be estimated not only by order but also by magnitude.

From formal numerical solutions of elliptic PDE's to the true ones

“…(A possible explanation of such a big difference between the triangular and the full methods may be that the relaxation scheme local error for the full scheme is of order h 12 , h 10 , h 8 for the values, the first and the second derivatives, respectively, while for the triangular scheme this error is of order h 12 , h 8 , h 4 . See Section 6 for results of symbolic calculations.…”