Generalized Multistep Predictor-Corrector Methods

Abstract

The order p which is obtainable with a stable k -step method in the numerical solution of y′ = f ( x , y ) is limited to p = k + 1 by the theorems of Dahlquist. In the present paper the customary schemes are modified by including the value of the derivative at one “nonstep point;” as usual, this value is gained from an explicit predictor. It is shown that the order of these generalized predictor-corrector methods is not subject to the above restrictions; stable k -step schemes with p = 2 k + 2 have been constructed for k ≤ 4. Furthermore it is proved that methods of order p actually converge like h p uniformly in a given interval of integration. Numerical examples give some first evidence of the power of the new methods.