A Crank–Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity

Abstract

A Crank–Nicolson-type finite-difference scheme is proposed and analyzed for a nonlinear partial integro-differential equation arising from viscoelasticity. The time derivative is approximated by the Crank–Nicolson scheme and the Riemann–Liouville fractional integral term is treated by means of the trapezoidal convolution quadrature rule. To construct a fully discrete difference scheme, the standard centered difference formula is utilized to approximate the second-order spatial derivative and the Galerkin method based on piecewise linear test functions is used to discrete the nonlinear convection term. Theoretical results of stability and convergence are derived using the non-negative character of the real quadratic form associated with the convolution quadrature, and combining with the discrete Gronwall inequality. Besides, a fixed point iterative numerical algorithm, which fills the gap that the existed numerical schemes have only theoretical analysis but no numerical results, is presented. Numerical results show the efficiency and feasibility of our scheme, and the orders of convergence are in good agreement with the theoretical results.