Computational method for singularly perturbed two-parameter parabolic convection-diffusion problems

Abstract

This paper deals with the numerical solution of singularly perturbed parabolic convection-diffusion problems with two small positive parameters multiplying the convection and diffusion terms. A parameter-uniform computational method is developed to solve these problems. The stability and consistency of the method are well established. Numerical experimentation is done and it is observed that the formulated method is stable, consistent and gives more accurate results than some methods exist in the literature.

Keywords:

• singularly perturbed
• parabolic convection-diffusion
• computational method
• two-parameter
• parameter uniform

PUBLIC INTEREST STATEMENT

Singularly perturbed differential equations are very important to describe some phenomena like viscous flow of fluids at high Reynolds number, convective heat transport problem, electromagnetic field problem in moving media, financial modeling, and turbulence model, reaction-diffusion process, etc. The solution to such problems changes unexpectedly in some regions of the domain where it exhibits boundary layers. The construction of numerical schemes which is not affected by the singular perturbation parameter is, however, not an easy task because of the singularity in the layer regions. But, the present work formulated a numerical method that is convergent independent of the effect of the singular perturbation parameter and is very helpful for scholars working in the aforementioned areas.

List of Symbols

T	=	Maximum time value
M	=	Number of subintervals of the time domain
N	=	Number of subintervals of the space domain
k	=	Stepsize in the time direction
h	=	Stepsize in the space direction
$D_t^M$	=	Time mesh (the set of time nodes)
$D_x^N$	=	Space mesh (the set of space nodes)
TE1	=	Truncation error in the time direction
TE2	=	Truncation error in the space direction
C	=	Any generic constant independent of the perturbation parameters and the step sizes
U	=	The approximate solution to the analytical solution u
$E_{\epsilon ,\mu }^{N,M}$	=	Maximum pointwise error
R	=	Rate of convergence
Greek letters	=
$\epsilon$	=	Perturbation parameters multiplying the diffusion term
$\mu$	=	Perturbation parameters multiplying the convection term
${\tau }_1$	=	Reminder term of the Taylor series expansion in the time direction
$\alpha$	=	The smallest positive value of the convection coefficient on the given domain
${\gamma }^2$	=	Denominator function which replaces h2
$\xi$	=	Amplification factor

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Gemechis File Duressa

The research of the authors spans both on pure and applied mathematics. Much of their work has been focused on numerical solution of singular perturbation problems; finite difference methods, finite element methods, fitted operator and fitted mesh methods for solving singularly perturbed boundary value problem in ODEs and PDEs.

Gemechis File Duressa is an Associate Professor of Mathematics and faculty member of the Department of Mathematics, College of Natural Sciences at Jimma University. So far he has published more than 50 research articles in reputable journals.

Tariku Birabasa Mekonnen is a Ph.D. Scholar at the Department of Mathematics, College of Natural and Computational Sciences, Wollega University, Nekemte, Ethiopia.