An exponentially fitted numerical technique for singularly perturbed Burgers-Fisher equation on a layer adapted mesh

ABSTRACT

The main aim of this paper is to propose a robust and efficient numerical scheme for solving time dependent singularly perturbed Burgers-Fisher problem based on mesh adaptive strategy in the finite element framework. The Burgers-Fisher equation in one space dimension is a partial differential equation which exhibits travelling wave phenomenon. Since Burgers-Fisher equation is a non-linear problem, quasilinearization process has been used to deal with nonlinearity occurring in the problem. Time discretization has been performed using implicit Euler method. Then spatial discretization has been carried out using finite element technique based on exponentially fitted splines on piecewise uniform Shishkin mesh. The stability of the proposed numerical scheme has been discussed. At the end, it has been shown numerically that the proposed method is very much effective for capturing sharp boundary layers arising in the solution as singular perturbation parameter $ϵ\to 0$.

KEYWORDS:

• Burgers-Fisher equation
• adaptive finite element technique
• quasilinearization
• implicit Euler method
• singularly perturbed problems
• boundary layer
• Shishkin mesh

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M50
• 65M

Disclosure statement

No potential conflict of interest was reported by the authors.