Interpolation for variable order, Runge-Kutta methods (Q1098579)

The problem of constructing a continuous solution of initial-value problems for a system of ordinary differential equations is considered. Extrapolation procedures based on Euler's method and defect correction using a one-step method are discussed from this point of view. From this discussion a new scheme, similar to extrapolation, based on repeated improvement of a simple starting polynomial approximation is developed. This can be used in a step-by-step fashion and may be regarded as a variable order Runge-Kutta method with an interpolation capability.