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.