Three semi-implicit compact finite difference schemes for the nonlinear partial integro-differential equation arising from viscoelasticity

ABSTRACT

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.

KEYWORDS:

• Nonlinear partial integro-differential equation
• compact finite difference scheme
• semi-implicit scheme
• convolution quadrature
• numerical experiment

Acknowledgments

The authors are grateful to the reviewers for their helpful and valuable suggestions for the improvement of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The project was supported by the Natural Science Foundation of Hunan Province [No.2018JJ2669], the Scientific Research Foundation of Hunan Provincial Education Department [No.17B277], the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering [No.2017TP1017], the Startup Scientific Research Foundation of CSUFT [No.2017YJ026].