A parameter-uniform Schwarz method for a singularly perturbed reaction–diffusion problem with an interior layer

Abstract

In this paper we consider numerical methods for a singularly perturbed reaction-diffusion problem with a discontinuous source term. We show that such a problem arises naturally in the context of models of simple semiconductor devices. We construct a numerical method consisting of a standard finite difference operator and a non-standard piecewise-uniform mesh. The mesh is fitted to the boundary and interior layers that occur in the solution of the problem. We show by extensive computations that, for this problem, this method is parameteruniform in the maximum norm, in the sense that the numerical solutions converge in the maximum norm uniformly with respect to the singular perturbation parameter. The semiconductor device equationsThe stationary behaviour of semiconductor devices can be described by the nonlinear system of second-order elliptic equations of van Roosbroeck [6]. In one dimension, for example, a simple p-n diode Ω = (0, 1) with two Ohmic contacts, an anode at x = 0 and a cathode at x = 1, is governed by the equationson Ω with appropriate boundary conditions at x = 0 and x = 1. Here ψ is the electrostatic potential, η and ρ are the electron and hole concentrations in the Slotboom variables, D denotes the doping function * Corresponding author.0168-9274/00/$ -see front matter © 2000 IMACS. Published by Elsevier Science B.V. All rights reserved PII: S 0168-9274(99 )00 1 40 -3

Solving Third-Order Singularly Perturbed Problems: Exponentially and Polynomially Fitted Trial Functions

“…Some authors (Valanarasu & Ramanujam, 2007;Miller, O'Riordan & Wang, 2000;Roos & Zarin, 2002;Farrel, Hegarty, Miller, O'Riordan & Shishkin, 2004) have considered the singularly perturbed problems with discontinuous source terms, which lead to a weak interior layer in the singular solution. It is interesting that the present weak-form integral equation method can handle this type problem without inducing any difficulty because the influence of the source term f (x) on the solution is given through the integral term…”

A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer

“…1 College of Finance and Trade, Ningbo Dahongying University, Ningbo, China. 2 Institute of Mathematics, Zhejiang Wanli University, Ningbo, China.…”

On An Equal Fourth-Order-Accurate Temporal/Spatial Scheme for the Convection-Diffusion Equation

“…In Table 1 the solution quality of the proposed equal-order finite-difference scheme is clearly shown to outperform the scheme of Miller et al [28].…”

“…It is well known that this problem has the interior layer developed at x ¼ 0:5 and the boundary layer formed at x ¼ 1 according to the solution derived as follows [28]:…”