A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters

ABSTRACT

In the present paper, a parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is proved to be uniformly convergent of order two in both the spatial and temporal variables. Numerical experiments support the theoretically proved higher order of convergence and show that the present scheme gives better accuracy and convergence compared of other existing methods in the literature.

Keywords:

• Singular perturbation
• two small parameters
• piecewise-uniform mesh
• finite difference
• Richardson extrapolation
• stability and convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N06
• 65N12
• 65N15

Disclosure statement

No potential conflict of interest was reported by the authors.