Difference Methods for Parabolic History Value Problems

This paper is concerned with one-step difference methods for parabolic history value problems in one space variable. These problems, which have the feature that the evolution of the solution is influenced by 'all its past’ occur in the theory of viscoelastic liquids (materials with 'memory’). The history dependence is represented by a Volterra-integral in the equation of motion. Using recently obtained existence results (see Renardy [9]) and smoothness assumptions on the solution we derive a local stability and convergence result for a Crank-Nicolson-type difference scheme by interpreting the linearized scheme as perturbation of a strictly parabolic scheme without memory term. Second order convergence is shown on sufficiently small time intervals. The presented approach carries over to other one-step difference methods like implicit and explicit Euler schemes.