High-order essentially non-oscillatory scheme for viscoelasticity with fading memory

Abstract

In this paper we describe the application of high-order essentially nonoscillatory (ENO) finite difference schemes to the viscoelastic model with fading memory. ENO schemes can capture shocks as well as various smooth structures in the solution to a high-order accuracy without spurious numerical oscillations. We first verify the stability and resolution of the scheme. We apply the scheme to a nonlinear problem with a known smooth solution and check the order of accuracy. Then we apply the scheme to a linear problem with initial discontinuities. Discontinuity locations and strengths in the solutions of such problems can be found explicitly by making use of a pointwise estimate obtained in this paper for the Green’s function of the equations, which contains two Dirac δ \delta -functions decaying exponentially. We check the resolution of the discontinuities by the scheme. After verifying that the scheme is indeed high-order accurate, produces sharp, non-oscillatory shocks with the correct location and strength, we then proceed in applying it to the nonlinear case with discontinuous or smooth initial conditions, and study the local properties (in time) as well as the long time behavior of the solutions. We conclude that the ENO scheme is a robust, accurate numerical tool to supplement theoretical analysis to study such equations with memory terms. It should also provide an efficient and reliable practical tool when such equations must be solved numerically in applications.