Alternative stepsize strategies for Adams predictor–corrector codes

Adams predictor–corrector methods are among the most widely used algorithms for solving initial value problems in ordinary differential equations. Adaptive stepsize techniques are employed to enhance the numerical stability and accuracy of these methods. This paper deals with the stepsize-control (SC) stability of Adams methods. The SC-stability conditions have been obtained for low-order Adams predictor–corrector methods using the standard stepsize strategies. However, when the stepsize is restricted by stability, it oscillates and frequent step rejections are observed. For the physically important case of real dominant eigenvalue of the Jacobian, the Adams methods are not SC-stable in general. In this paper, we investigate alternative stepsize strategies to smooth out the stepsize behaviour. It has been found that PI stepsize controller and estimation techniques, which have been developed for Runge–Kutta methods, fail to give good results in the case of Adams methods. A combined strategy has been formulated which eliminates stepsize oscillations and results in a smooth stepsize behaviour. All programming has been carried out in the integrated environment of standard software package MATLAB©.