HIGHLY CONTINUOUS INTERPOLANTS FOR ONE-STEP ODE SOLVERS AND THEIR APPLICATION TO RUNGE-KUTTA METHODS

We suggest a general method fur 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 constructi on of these interpolants one uses, along with the numerical data of th e discrete solution of a problem provided by a typical one-step method at endstep points, high-order derivative approximations of this solut ion. This approach has two main advantages. It allows an easy way of c onstruction of high-order Runge-Kutta and Nystrom interpolants with re duced cost in additional function evaluations that also preserve the o ne-step nature of the underlying discrete ODE solver. Moreover. for pr oblems which are known to possess a solution of high smoothness, the a pproximating interpolant resembles this characteristic, a property tha t on occasion might be desirable. An analysis of the stability behavio r 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 deriva tives 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 c ases, 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 promis ing) types of methods, as they are the general linear methods for firs t- and second-order differential Equations. Many particular variants o f the new method for first-order differential equations that have good prospects of Ending a practical implementation are fully analyzed wit h respect to their stability characteristics. A detailed application c oncerning the construction of C-2 and C-3 continuous extensions for so me fifth- and sixth-order Runge-Kutta pairs. supplemented by a detaile d study of tile local truncation error characteristics of a class of i nterpolants of this type is also provided. Various numerical examples show, in these cases, several advantages of the newly proposed techniq ue with respect to function evaluation cost and global error behavior in comparison with others currently in use.