A hybrid computational scheme for singularly perturbed Burgers'-Huxley equation

Abstract

This paper aims to construct and analyze a hybrid computational method for the nonlinear singularly perturbed Burgers'-Huxley equation. The presence of the perturbation parameter and non-linearity in the considered problem makes it difficult to solve the problem analytically and using classical numerical techniques on uniform step sizes as $\epsilon$ goes small. To elucidate such limitations, one can rely on non-classical numerical techniques. In this paper, one such parameter uniform numerical method is designed for the considered problem. The method is constructed: •Firstly, the non-linear terms are linearized via Newton-Raphson-Kantorovich technique. The linearized problem is discretized by the implicit Euler method in the temporal direction.•Secondly, the obtained equation is solved by employing the hybrid computational method comprised of the cubic spline in tension method in the inner layer region and the midpoint upwind method in the outer layer region on a piecewise uniform Shishkin mesh.•Finally, an error analysis of the method is done and observed that the proposed method is parameter uniform convergent with the order of convergence $O(\tau +N^{-2}{\ln }^{3}N)$. Three examples are presented and the results are compared to some existing schemes in the literature to demonstrate the reliability of the proposed scheme.

Keywords: A Hybrid Computational Scheme for Singularly Perturbed Burger-Huxley Equation; Burger Huxley equation; Cubic-spline in tension method; Implicit Euler method; Nonlinear problem.