Pointwise uniformly convergent numerical treatment for the non-stationary Burger–Huxley equation using grid equidistribution

Abstract

This article presents a numerical method for solving the singularly perturbed Burger–Huxley equation on a rectangular domain. That is, the highest-order derivative term in the equation is multiplied by a very small parameter. This small parameter is known as the perturbation parameter. When the perturbation parameter specifying the problem tends to zero, the solution of the perturbed problem exhibits layer behaviour in the outflow boundary region. Most conventional methods fail to capture this layer behaviour. For this reason, there is much current interest in the development of a robust numerical method that may handle the difficulties occurring due to the presence of the perturbation parameter and the nonlinearity of the problem. To solve both of these difficulties a numerical method is constructed. The first step in this direction is the discretization of the time variable using Euler's implicit method with a constant time step. This produces a nonlinear stationary singularly perturbed semidiscrete problem class. The problem class is then linearized using the quasilinearization process. This is followed by discretization in space, which uses the standard upwind finite difference operator. An extensive amount of analysis is carried out in order to establish the convergence and stability of the proposed method. Numerical experiments are carried out for model problems to illustrate graphically the theoretical results. The results indicate that the scheme faithfully mimics the dynamics of the differential equation.