Convergence of linear multistep methods for differential equations with discontinuities

Summary

A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [1] we prove that a linear multistep method of design orderq≧p≧1 which satisfies the weak stability root condition, applied to the differential equationy′ (t)=f (t, y (t)) wheref is Lipschitz continuous in its second argument, will exhibit actual convergence of ordero(h ^p−1) ify has a (p−1)th derivativey ^(p−1) that is a Riemann integral and ordero(h ^p) ify ^(p−1) is the integral of a function of bounded variation. This result applies for a functiony taking on values in any real vector space, finite or infinite dimensional.