Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection–diffusion equations

Abstract

In this paper, we discuss the parameter-uniform finite difference method for a coupled system of singularly perturbed convection-diffusion equations. The leading term of each equation is multiplied by a small but different magnitude positive parameter, which leads to the overlap and interact boundary layer. We analyze the boundary layer and construct a piecewise-uniform mesh on the variant of the Shishkin mesh. We prove that our schemes converge almost first-order uniformly with respect to small parameters. We present some numerical experiments to support our theoretical analysis.

Solving Systems of Singularly Perturbed Convection Diffusion Problems via Initial Value Method

“…Theorem 6. Let 푦(푥) be the solution of (1a)-(2b) and 푢 0 (푥) be its reduced problem solution defined by (17). en…”

“…In the papers [15,16], a class of strongly coupled systems of singularly perturbed convection-diffusion equations are examined. e scholars in [17][18][19][20][21], considered weakly coupled systems of singularly perturbed convection-diffusion equations with equal or different diffusion parameters. A brief survey of article on the current progress about the numerical treatment of systems of singularly perturbed differential equations is also discussed in [22].…”

Uniformly-Convergent Numerical Methods for a System of Coupled Singularly Perturbed Convection–diffusion Equations With Mixed Type Boundary Conditions

“…Most of this work has concentrated on problems involving a single differential equation. Only a few authors have developed robust parameteruniform numerical methods for system of singularly perturbed ordinary differential equations (see [2,4,8,9,10,11,15,16,19] and references therein). While many finite difference methods have been proposed to approximate such solutions, there has been much less research into the finite difference approximations of their derivatives, even though such approximations are desirable in certain applications (flux or drag).…”

Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations

“…Bellew and O'Riordan [4], Cen [5], Amiraliyev [6] and Andreev [7] used the finite difference method for a coupled system of two singularly perturbed convectiondiffusion equations.…”