Three Semi-Implicit Compact Finite Difference Schemes For The Nonlinear Partial Integro-Differential Equation Arising From Viscoelasticity

Three semi-implicit compact finite difference schemes are described for the nonlinear partial integro-differential equation arising from viscoelasticity. In our methods, the time derivative is approximated by the backward-Euler, Crank-Nicolson-type and formally second-order backward differentiation formula scheme, respectively, and the convolution quadrature formula is employed to process the Riemann-Liouville fractional integral term. Fully discrete difference schemes are constructed with the spatial discretization by the compact finite difference formula. Meanwhile, the semi-implicit technique is used to deal with nonlinear convection term uu(x) . In the numerical experiment, the comparisons between our methods and existing methods demonstrate the efficiency of our methods.