Lax-Wendroff Methods for Hyperbolic History Value Problems.

The motion of viscoelastic materials can be modelled by partial integrodifferential equations. For several model problems, recent investigations have been concerned with the question whether or not these equations allow the development of shocks. This paper is concerned with Lax-Wendroff methods for a class of hyperbolic history value problems. These problems have the feature that globally in time smooth solutions exist if the data are sufficiently small and that solutions develop singularities for large data. The authors prove second order convergence of the Lax-Wendroff method for smooth solutions and investigate numerically the dependence on the initial data. They demonstrate the occurrence of shock type singularities and compare the results to the quasilinear wave equation without Volterra term.