### Abstract:

A class of numerical methods for solving nonstiff initial value problems in ordinary
differential equations has been developed. These methods, designated RKrGLn, are
based on a Runge-Kutta method of order r (RKr), and Gauss-Legendre integration
over n+ 1 nodes. The interval of integration for the initial value problem is subdivided
into an integer number of subintervals. On each of these n + 1 nodes are defined in
accordance with the zeros of the Legendre polynomial of degree n. The Runge-Kutta
method is used to find an approximate solution at each of these nodes; Gauss-Legendre
integration is used to find the solution at the endpoint of the subinterval. The process
then carries over to the next subinterval.
We find that for a suitable choice of n, the order of the local error of the Runge-
Kutta method (r + 1) is preserved in the global error of RKrGLn. However, a poor
choice of n can actually limit the order of RKrGLn, irrespective of the choice of r.
What is more, the inclusion of Gauss-Legendre integration slightly reduces the number
of arithmetical operations required to find a solution, in comparison with RKr at
the same number of nodes. These two factors combine to ensure that RKrGLn is
considerably more efficient than RKr, particularly when very accurate solutions are
sought.
Attempts to control the error in RKrGLn have been made. The local error has
been successfully controlled using a variable stepsize strategy, similar to that generally
used in RK methods. The difference lies in that it is the size of each subinterval that
is controlled in RKrGLn, rather than each individual stepsize. Nevertheless, local
error has been successfully controlled for relative tolerances ranging from 10 -4 to
10-10 . We have also developed algorithms for estimating and controlling the global
error. These algorithms require that a complete solution be obtained for a specified
distribution of nodes, after which the global error is estimated and then, if necessary,
a new node distribution is determined and another solution obtained. The algorithms
are based on Richardson extrapolation and the use of low-order and high-order pairs.
The algorithms have successfully achieved desired relative global errors as small as
10-1° .
We have briefly studied how RKrGLn may be used to solve stiff systems. We have
determined the intervals of stability for several RKrGLn methods on the real line, and
used this to develop an algorithm to solve a stiff problem. The algorithm is based on
the idea of stepsize/subinterval adjustment, and has been used to successfully solve
the van der Pol system.
Lagrange interpolation on each subinterval has been implemented to obtain a piecewise
continuous polynomial approximation to the numerical solution, with same order
error, which can be used to find the solution at arbitrary nodes.