Stability of linear multistep methods and applications to nonlinear parabolic problems

Abstract

In the present paper, stability and convergence properties of linear multistep methods are investigated. The attention is focused on parabolic problems and variable stepsizes. Under weak assumptions on the method and the stepsize sequence an asymptotic stability result is shown. Further, stability bounds for linear nonautonomous parabolic problems with Hölder continuous operator are given. With the help of these results, convergence estimates for semilinear and fully nonlinear parabolic problems are derived.