A class of nonstandard finite difference methods for the Burgers–Huxley equation

In this work, a class of nonstandard finite difference (NSFD) methods is proposed to approximate the exact solution of a diffusive partial differential equation with Burgers convection effect and Huxley reaction law. The model under investigation has a positive and bounded solution if the initial and boundary conditions satisfy some restrictions, and some travelling-wave solutions, which are positive, bounded and monotone in both space and time. It is established that the proposed methods are capable of preserving the positivity, boundedness and monotonicity of the exact solution under some conditions on the step sizes. In addition, the numerical stability and convergence of these methods are analysed. Finally, some numerical examples are given to verify the correctness of the theoretical results and the validity of our methods.