Numerical Solution of Systems of Singularly Perturbed Differential Equations

Abstract

A survey is given of current research into the numerical solution of timeindependent systems of second-order differential equations whose diffusion coefficients are small parameters. Such problems are in general singularly perturbed. The equations in these systems may be coupled through their reaction and/or convection terms. Only numerical methods whose accuracy is guaranteed for all values of the diffusion parameters are considered here. Some new unifying results are also presented.2000 Mathematics Subject Classification: 65-02, 65L10, 65L12, 65L50, 65M06, 65M50.

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).…”

An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction–diffusion equations

“…In recent years there has been a considerable interest in coupled system of singularly perturbed problems. A brief survey of numerical methods developed for these problems is given in [5]. Methods of high order convergence reduce the computational cost to find good numerical approximations.…”

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

“…), that is, no matter how small ϵ is, we always have under some appropriate conditions, where C is a positive constant independent of ϵ and h . Although the IAS scheme is demonstrated to be efficient for obtaining stable and accurate numerical results for scalar singularly perturbed convection‐diffusion equations, to the best of our knowledge, it has never been achieved before in the literature to successfully construct an effective IAS scheme for solving the strongly coupled system (1.1) of singularly perturbed convection‐diffusion equations . Very recently, we proposed in a novel technique to apply the IAS scheme for scalar equations to derive a formally second‐order scheme for the strongly coupled system (1.1).…”

“…However, it is often difficult to resolve numerically the high gradients near the layer regions. Consequently, this simple coupled system (1.1) has been the focus of intense research for quite some time, see for example, [10][11][12][13][14][15][16][17][18][19][20] and references cited therein.…”