Convergence and Stability of Multistep Methods Solving Nonlinear Initial Value Problems

Abstract

Convergence and stability of initial and boundary value multistep methods are analyzed for a class of nonlinear problems, satisfying a one-sided Lipschitz condition. The linear multistep methods are recast to handle the numerical solution globally on the time interval. This allows us to use the theory of Toeplitz matrices to show that, under suitable assumptions, global properties of the exact solution are preserved by its numerical approximation. In particular, a new concept of stability, which avoids the unpleasant passage to one-leg methods, is introduced.