An improved starting step of the G-B-S-method for the solution of ordinary differential equations (Q1114350)

This paper proposes a modification of the starting step when the midpoint rule is applied to extrapolation methods in the numerical solution of ordinary differential equations. In traditional extrapolation procedures [cf. \textit{R. Bulirsch} and \textit{J. Stoer}, Numer. Math. 8, 1-13 (1966; Zbl 0135.379)], \textit{P. Deuflhard} [Preprint Univ. Heidelberg (1980)] the additional approximation required to apply the midpoint rule is computed by Euler's method and a final smoothing procedure is included to improve the stability of the method. Here the final smoothing has been skipped and the Euler step is replaced by \(y_ 1=y_ 0+hf(x_ 0,y_ 0)+(h^ 2/2)c(x_ 0,y_ 0),\) where \(c(x_ 0,y_ 0)\) is an approximation to \(y''(x_ 0)\) given by \(H^{-1}[f(x_ 0+H,y_ 0+Hf(x_ 0,y_ 0)-f(x_ 0,y_ 0)].\) The aim of this new starting step is to reduce the oscillating terms in the \(h^ 2\)-expansion of the error. The paper ends comparing the extrapolation code DIFEX1 (due to Deuflhard) with two modified versions of it for the test problems of DETEST. It is shown that, with the modified code, an average saving of about 8 \% in the total number of function calls is obtained.