On the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation

Abstract

In this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19 (2014), pp. 1907–1920]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some numerical experiments are conducted in order to assess the validity of the analytical results. We conclude that the methodology under investigation is a fast, nonlinear, explicit, stable, convergent numerical technique that preserves the positivity, the boundedness and the monotonicity of approximations, making it an ideal tool in the study of some travelling-wave solutions of the mathematical model of interest. This note closes proposing new avenues of future research.

Keywords::

• finite-difference scheme
• nonlinear method
• convergence analysis
• generalized Burgers–Fisher model

AMS Subject Classification::

• 65Q10
• 65M06
• 65M12
• 35K55