Singularly Perturbed Problems Modeling Reaction-convection-diffusion Processes

Abstract

In this paper, parameter -uniform numerical methods for singularly perturbed ordinary differential equations containing two small parameters are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of numerical approximations.2000 Mathematics Subject Classification: 65L10, 65L12, 65L20.

Convection‐diffusion‐reaction problems on a B‐type mesh

“…Numerical methods for one-dimensional singularly perturbed problems with two small parameters are considered in [3,4,[6][7][8][10][11][12], but on a Shishkin-type mesh. Vulanović [12] considered Shishkin and Bakhvalov meshes but assumed ε 2 = ε p+1/2 1 with p > 0.…”

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

“…Vulanović [17] considers finite difference methods in the case of µ = ε 1 2 +λ , λ > 0. More recently, parameter-uniform numerical methods for the steady-state version of (1.1) were examined by Linß and Roos [4], Roos and Uzelac [11] and O'Riordan et al [9]. Both [4] and [9] are concerned with finite difference methods and apply standard finite difference operators on special piecewise uniform meshes.…”

“…More recently, parameter-uniform numerical methods for the steady-state version of (1.1) were examined by Linß and Roos [4], Roos and Uzelac [11] and O'Riordan et al [9]. Both [4] and [9] are concerned with finite difference methods and apply standard finite difference operators on special piecewise uniform meshes. In [11] the problem is solved using the streamline-diffusion finite element method on a piecewise uniform mesh.…”

“…We apply similar analytical techniques to those used in [9] for a singularly perturbed ordinary differential equation with two small parameters. The argument consists of establishing a maximum principle, a decomposition of the solution into regular and layer components and deriving sharp parameter-explicit bounds on these components and their derivatives.…”

“…The discrete solution is decomposed into analogous components and the numerical error between the discrete and continuous components are analysed separately using discrete maximum principle, truncation error analysis, and appropriate barrier functions. In [9], the piecewise uniform mesh constructed consisted of the two transition points…”