A two-step conditionally bounded numerical integrator to approximate some traveling-wave solutions of a diffusion-reaction equation

Abstract

We develop a finite-difference scheme to approximate the bounded solutions of the classical Fisher–Kolmogorov–Petrovsky–Piskunov equation from population dynamics, in which the nonlinear reaction term assumes a generalized logistic form. Historically, the existence of wave-front solutions for this model is a well-known fact; more generally, the existence of solutions of this equation which are bounded between $0$ and $1$ at all time, is likewise known, whence the need to develop numerical methods that guarantee the positivity and the boundedness of such solutions follows necessarily. The method is implicit, relatively easy to implement, and is capable of preserving the positivity and the boundedness of the new approximations under a simple parameter constraint. The proof of the most important properties of the scheme is carried out with the help of the theory of $M$-matrices. Finally, the technique is tested against some traveling-wave solutions of the model under investigation; the results evince the fact that the method performs well in the cases considered.