On some explicit non-standard methods to approximate nonnegative solutions of a weakly hyperbolic equation with logistic nonlinearity

Abstract

We introduce non-standard, finite-difference schemes to approximate nonnegative solutions of a weakly hyperbolic (that is, a hyperbolic partial differential equation in which the second-order time-derivative is multiplied by a relatively small positive constant), nonlinear partial differential equation that generalizes the well-known equation of Fisher-KPP from mathematical biology. The methods are consistent of order 𝒪(Δ t+(Δ x)²). As a means to verify the validity of the techniques, we compare our numerical simulations with known exact solutions of particular cases of our model. The results show that there is an excellent agreement between the theory and the computational outcomes.

Keywords:

• finite-difference scheme
• non-standard method
• positivity
• nonlinear reaction
• logistic nonlinearity

2000 AMS Subject Classifications :

• 65M06
• 35L20
• 35L70
• 92B05

Acknowledgements

One of the author (J.E.M.D.) wishes to express his gratitude to Dr F. J. Álvarez Rodríguez, dean of the Faculty of Sciences of the Universidad Autónoma de Aguascalientes, and to Dr F. J. Avelar González, director of the Office for Research of the same university, for providing the computational resources necessary to produce this work. He also wishes to thank the Mexican Academy of Sciences and the Foundation Mexico–US for the Sciences for supporting this research project. The present article summarizes results of project PIM09-1 of the Universidad Autónoma de Aguascalientes, and it was completed when J.E.M.D. visited the University of New Orleans in the Fall of 2009. Finally, JEMD wishes to take this opportunity to acknowledge the kindness and the hospitality he enjoyed in this university.