ABSTRACT
This chapter introduces the multistep processes. Some simple formulae will be constructed from Taylor expansions of the local truncation error, the same method in principle as that employed in the Runge–Kutta (RK) case. Unlike RK processes, which up to now have been explicit, it is normal to apply implicit multistep schemes for general problems. In the multistep case, this determination is much simpler because only total derivatives need to be considered, instead of the large number of elementary differentials which arise from Runge-Kutta expansions. In the multistep case, only one extra condition needs to be imposed to raise the order by one, considerably simpler than the RK alternative. Coding the Predictor–Corrector algorithm in Fortran 90 is quite straightforward. The program is a little more complicated than the RK equivalent because a starting mechanism has to be provided.
