Cyclic Composite Multistep Predictor-Corrector Methods

Multistep predictor-corrector methods are commonly used for the numerical solution of ordinary differential equations. In its simplest form, a k-step method with accuracy of order exceeding $k+2$ is unstable. Methods such as those of Gragg and Stetter [7] and of Butcher [1] obtain high accuracy while retaining stability. However, the price paid is additional evaluation(s) of the function $f(x,y)$ occurring in the differential equation $y' = f(x,y)$. Also, there is an upper bound on the step number k. For $k > 7$ stable methods of the Gragg and Stetter or Butcher type do not exist. In this paper, we consider composite methods using M different correctors applied cyclically. Methods of this type were first discussed in [8]. Composite methods of this type may be considered from the standpoint of difference equations in M-dimensional space. We show that a composite method with accuracy of order $2k - 1$ can be stable and entails no additional computation. A small amount of extra programming is required since M correctors must be stored instead of a single corrector. We also prove the convergence of our difference scheme under mild assumptions on the function , $f(x,y)$. Finally, we exhibit the coefficients of several composite methods for $k = 2,3,4$ and tabulate the results for several numerical test cases.