Multistep Methods and General Linear Methods

Abstract

This chapter is devoted to the study of multistep and general multivalue methods. After retracing their historical development (Adams, Nyström, Milne, BDF) we study in the subsequent sections the order, stability and convergence properties of these methods. Convergence is most elegantly set in the framework of onestep methods in higher dimensions. Sections III.5 and III.6 are devoted to variable step size and Nordsieck methods. We then discuss the various available codes and compare them on the numerical examples of Section II.10 as well as on some equations of high dimension. Before closing the chapter with a section on special methods for second order equations, we discuss two highly theoretical subjects: one on general linear methods, including Runge-Kutta methods as well as multistep methods and many generalizations, and the other on the asymptotic expansion of the global error of such methods.