Convergence of linear multistep methods for differential equations with discontinuities

SummaryA 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(hp−1) ify has a (p−1)th derivativey(p−1) that is a Riemann integral and ordero(hp) 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.