Optimal Predictor-Corrector Method for Systems of Second-Order Differential Equations

A very general class of predictor-corrector schemes for numerically integrating a system of equations of the type x + f(x, t) = 0 is examined. This class includes the schemes obtained by combining one explicit predictor with p implicit correctors (allowing the use of a different corrector for each evaluation). Both the predictor and the correctors are assumed to be two- or three-step operators. It is shown that all of the schemes within this class are conditionally stable, that is, are stable only if the time increment A/ does not exceed a critical value. Furthermore, it is shown that the central difference scheme is the optimal one in the sense that for all the schemes within the above class, the value of Ar divided by the number of evaluations of f(x, t) is never greater than for the central difference one. The same conclusions are valid for any scheme with a characteristic equation of the type p3+ d^p2 + d 2p + d3 = 0, where dk are polynomials in (coAr)2, where co is the largest natural frequency of the system.