A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady Burger–Huxley equation

Abstract

In this paper, we present a numerical method which deals with the solvability of initial boundary value problem and related general properties of singularly perturbed Burger–Huxley equation. Singular perturbation problems are the problems in which a very small positive parameter is multiplied to the highest order derivative term. When this parameter approaches the limiting value of interest, i.e. ϵ → 0 , the problem exhibits boundary layers and most of the conventional methods fails to capture this effect. Thus, the quest for some new numerical techniques that may handle the difficulties occurring due to the presence of perturbation parameter and nonlinearity in the problem earns relevance. In this paper, one such parameter robust numerical scheme is constructed. The first step in this direction is the discretization of the time variable using the Euler implicit method with a constant time step. This produces a set of nonlinear stationary singularly perturbed semidiscrete problem class which is linearized further using the quasi-linearization process followed by spatial discretization using upwind finite difference operator on a piecewise uniform mesh. An extensive amount of analysis is carried out that uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical experiments is carried out in support of the predicted theory which validates the theoretical results computationally.