A linear multistep numerical integration scheme for solving systems of ordinary differential equations with oscillatory solutions

AbstractIn [1], a set of convergent and stable two-point formulae for obtaining the numerical solution of ordinary differential equations having oscillatory solutions was formulated. The derivation of these formulae was based on a non-polynomial interpolant which required the prior analytic evaluation of the higher order derivatives of the system before proceeding to the solution. In this paper, we present a linear multistep scheme of order four which circumvents this (often tedious) initial preparation. The necessary starting values for the integration scheme are generated by an adaptation of the variable order Gragg-Bulirsch-Stoer algorithm as formulated in [2]

Numerical treatment of singular initial value problems

AbstractAn efficient extrapolation scheme whose basic integrator is the inverse Euler scheme (cf. Fatunla, 1982) is proposed for nonlinear singular initial value problems y′ = ƒ(x, y), y(0) = y0. The automatic (polynomial/rational) extrapolation code DIFEXI (Deufhard 1983, 1985) is modified to accomodate the basic integrator. The new algorithm was implemented in variable step, variable order mode and compares favourably with the earlier works