A second-order linearized finite difference scheme for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation

Abstract

The purpose of this paper is to study the numerical simulation of the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. After introducing a new variable, the integro-differential equation is transformed into an equivalent coupled system of first-order differential equations. A second-order accurate difference scheme is constructed for the new system of equations, which is proved to be local uncoupled by separation of variables. It is also proved that the scheme is uniquely solvable and second-order convergent in both time and space in L ₂-norm. A numerical example is given to demonstrate the theoretical results.

Keywords:

• generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation
• difference scheme
• local uncoupled
• uniquely solvable
• convergent

2010 AMS Subject Classifications :

• 65M06
• 65M12
• 65M15
• 65R10

Acknowledgements

We thank deeply the referees for their valuable suggestions. The work was supported by Natural Science Foundation of China (Grant Number 10871044).