Implicit single step methods by spline-like functions for solution of ordinary differential equations (Q1118362)

A family of one step implicit algorithms is given which produces continuously differentiable approximations to a first order ordinary differential equation. The approximations are not splines but are spline- like. Let \((i)\quad y(x+h)=y(x)+hg(y,h,f)\) determine an explicit one step p-th order approximation to the solution y(x) of \(y'=f(x,y)\) at nodes \(x_ n=x_ 0+nh\), \(n=0,1,..\). The new algorithm is obtained by replacing (i) by \((ii)\quad y(x+h) = y(x)+hg(y,h,f)+wh^{p+1}\) where \(w\) is chosen so that \(y'(x+h)=f(x+h,y(x+h)).\) It is proved that (ii) is a \((p+1)\)-th order implicit algorithm. When (i) is specialized to be the Euler formula (ii) turns out to be \(A\)-stable. A computer program has been written in which (i) is chosen to be a Runge-Kutta algorithm of order \(p=5\) with continuously variable weights due to Sarafyan [cf. \textit{C. Outlaw}, \textit{L. Derr} and \textit{D.Sarafyan}, ibid. 12A(6), 815-824 (1986; Zbl 0626.65069)]. Two numerical examples are provided.