On the numerical integration of nonlinear initial value problems by linear multistep methods

We study the numerical solution of the nonlinear initial value problem $$left{ {begin{array}{*{20}c} {{{du(t)} mathord{left/ {vphantom {{du(t)} {dt}}} right. kern-nulldelimiterspace} {dt}} + Au(t) = f(t),t > 0} {u(0) = c,} end{array} } right.$$ whereA is a nonlinear operator in a real Hilbert space. The problem is discretized using linear multistep methods, and we assume that their stability regions have nonempty interiors. We give sharp bounds for the global error by relating the stability region of the method to the monotonicity properties ofA. In particular we study the case whereAu is the gradient of a convex functional φ(u).