A numerical scheme for a weakly coupled system of singularly perturbed delay differential equations on an adaptive mesh

Abstract

Summary In this paper, an adaptive mesh selection strategy is presented for solving a weakly coupled system of singularly perturbed delay differential equations of convection‐diffusion type using second order central finite difference scheme. Layer adaptive mesh is generated via an entropy production operator. The details of the location and width of the boundary layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. The method is independent of perturbation parameter and gives us an oscillation‐free solution. The applicability of the proposed method is illustrated by means of three examples.

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

“…In the literature, SPDDEs have been investigated widely by many authors and different numerical methods have been introduced. These include: Reproducing kernel method [16,28,29], initial value technique [59], numerical integration method [27,50,57], Numerov method [15], the method of hybrid difference schemes [13,14], discontinuous Galerkin method [64], collocation methods [39,60,63], Ritz-Galerkin method [34], hp−finite element method [46], fitted mesh technique [33,42], domain decomposition approach [56], cubic spline methods [36], finite difference methods [5,7,9,22,53] and so on [11,25,44,48,49,54].…”

Mesh selection strategies of the code TOM for Boundary Value Problems

“…In particular we recall the Shishkin mesh and the Bakhvalov mesh strategies, that have special properties of uniform convergence [6,7]. These strategies are effective when the location and the width of the layer are known, even if recently, adaptive meshes for weakly coupled system of singularly perturbed delay differential equations have been proposed in [8]. Robust hybrid methods of higher order for singularly perturbed convection-diffusion problems have been proposed in [9].…”