Numerical solution of neutral functional differential equations by Adams methods in divided difference form

A variable step and variable order algorithm for the numerical solution of neutral functional differential equations is described. This general purpose algorithm is based on the variable step formulation of the Adams methods represented in divided difference form. The Adams–Bashforth and Adams–Moulton methods are implemented in predictor–corrector mode. The detection of derivative discontinuities relies on the estimates of the local discretization errors. The restarting of the integration at each discontinuity point relies on the step size and order changing strategy based on the estimates of the local discretization errors. This algorithm reduces the overhead cost by implementing the Adams methods in divided difference form. This algorithm increases the reliability and efficiency by obtaining asymptotically correct estimates of the local discretization errors for the lower adjacent orders without any extra function evaluation by using local extrapolation. The results from three test examples are presented.