Approximation of derivative in a system of singularly perturbed convection-diffusion equations

Abstract

In this paper, a numerical method for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a small parameter multiplying the highest derivative is presented. Parameteruniform error bounds for the numerical solution and also to numerical derivative are established. Numerical results are provided to illustrate the theoretical results.

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

“…We split the argument into two cases depending on the localization of the mesh point. In the first case x i ∈ Ω N ∩ [σ, 1], using the arguments in [1] and [14,Lemma 6], for…”

“…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).…”

“…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). As far as author's knowledge goes only few works are reported in the literature (see [7,13,14] and references therein) for finding approximations to scaled derivatives of the solution for problems involving singularly perturbed second order ordinary differential equations with smooth/non-smooth data.…”

“…It may be noted that the source term and convection coefficient are smooth for the problem considered in [7]. Mythili Priyadharshini et al [14], have determined an estimate for the scaled derivative in the boundary layer region and non-scaled derivative in the outer region for the system of singularly perturbed convection-diffusion equations with Dirichlet boundary conditions. In [13], the authors have estimated the scaled derivative for a singularly perturbed second order ordinary differential equation with a discontinuous convection coefficient using a hybrid difference scheme.…”

Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs

“…Recently, the uniform first-order convergence of normalized flux by the upwind finite difference scheme using grid equidistribution was proved in [18]. As well known, the attainment of high accuracy in a numerical solutions does not automatically lead to good approximation of derivatives of exact solutions for convection-diffusion problems, and approximations of derivatives are desirable in many fields [22,18,14,19].…”