How many k-step linear block methods exist and which of them is the most efficient and simplest one?

摘要

There have appeared in the literature a lot of k-step block methods for solving initial-value problems. The methods consist in a set of k simultaneous multistep formulas over k non-overlapping intervals. A feature of block methods is that there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge–Kutta methods). All the formulas are usually obtained from a continuous approximation derived via interpolation and collocation at k+1 points. Nevertheless, all the k-step block methods thus obtained may be considered as different formulations of one of them, which results to be the most efficient and simple formulation of all of them. The theoretical analysis and the numerical experiments presented support this claim.