Highly Continuous Interpolants for One-step Ode Solvers and Their Application to Runge–kutta Methods∗

We suggest a general method for the construction of highly continuous interpolants for one-step methods applied to the numerical solution of initial value problems of ODEs of arbitrary order. For the construction of these interpolants one uses, along with the numerical data of the discrete solution of a problem provided by a typical one-step method at endstep points, high-order derivative approximations of this solution. This approach has two main advantages. It allows an easy way of construction of high-order Runge–Kutta and Nystrom interpolants with reduced cost in additional function evaluations that also preserve the one-step nature of the underlying discrete ODE solver. Moreover, for problems which are known to possess a solution of high smoothness, the approximating interpolant resembles this characteristic, a property that on occasion might be desirable. An analysis of the stability behavior of such interpolatory processes is carried out in the general case. A new numerical technique concerning the accurate determination of the stability behavior of numerical schemes involving higher order derivatives and/or approximations of the solution from previous grid-points over nonequidistant meshes is presented. This technique actually turns out to be of a wider interest, as it allows us to infer, in certain cases, more accurate results concerning the stability of, for example, the BDF formulas over variable stepsize grids. Moreover it may be used as a framework for analyzing more complex (and supposedly more promising) types of methods, as they are the general linear methods for firstand second-order differential equations. Many particular variants of the new method for first-order differential equations that have good prospects of finding a practical implementation are fully analyzed with respect to their stability characteristics. A detailed application concerning the construction of C2 and C3 continuous extensions for some fifthand sixth-order Runge–Kutta pairs, supplemented by a detailed study of the local truncation error characteristics of a class of interpolants of this type, is also provided. Various numerical examples show, in these cases, several advantages of the newly proposed technique with respect to function evaluation cost and global error behavior, in comparison with others currently in use.