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 orderqp1 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.This work was supported by Grant GJ-938 from the National Science Foundation