A convergent and dynamically consistent finite-difference method to approximate the positive and bounded solutions of the classical Burgers–Fisher equation

Abstract

In this work, we investigate both analytically and numerically a spatially two-dimensional advection–diffusion–reaction equation that generalizes the Burgers’ and the Fisher’s equations. The partial differential equation of interest is a nonlinear model for which the existence and the uniqueness of positive and bounded solutions are analytically established here. At the same time, we propose an exact finite-difference discretization of the Burgers–Fisher model of interest and show that, as the continuous counterpart, the method proposed is capable of preserving the positivity and the boundedness of the numerical approximations as well as the temporal and spatial monotonicity of the discrete initial–boundary conditions. It is shown that the method is convergent with first order in time and second order in space. We provide some simulations that illustrate the fact that the proposed technique preserves the positivity, the boundedness and the monotonicity.