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Uniform SecondOrder Difference Method for a Singularly Perturbed ThreePoint Boundary Value Problem
Advances in Difference Equations volume 2010, Article number: 102484 (2010)
Abstract
We consider a singularly perturbed onedimensional convectiondiffusion threepoint boundary value problem with zerothorder reduced equation. The monotone operator is combined with the piecewise uniform Shishkintype meshes. We show that the scheme is secondorder convergent, in the discrete maximum norm, independently of the perturbation parameter except for a logarithmic factor. Numerical examples support the theoretical results.
1. Introduction
We consider the following singularly perturbed threepoint boundary value problem:
where is the perturbation parameter, and, , and are given constants. The functions , and are sufficiently smooth. For the function has in general boundary layers at and .
Equations of this type arise in mathematical problems in many areas of mechanics and physics. Among these are the NavierStokes equations of fluid flow at high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, shear in secondorder fluids, control theory, electrical networks, and other physical models [1, 2].
Differential equations with a small parameter multiplying the highest order derivatives are called singularly perturbed differential equations. Typically, the solutions of such equations have steep gradients in narrow layer regions of the domain. Classical numerical methods are inappropriate for singularly perturbed problems. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value ; that is, methods that are convergence uniformly [1–5]. One of the simplest ways to derive such methods consists of using a class of special piecewise uniform meshes (a Shishkin mesh), (see, e.g., [4–8] for motivation for this type of mesh), which are constructed a priori in function of sizes of parameter , the problem data, and the number of corresponding mesh points.
Threepoint boundary value problems have been studied extensively in the literature. For a discussion of existence and uniqueness results and for applications of threepoint problems, see [9–12] and the references cited in them. Some approaches to approximating this type of problem have also been considered [13, 14]. However, the algorithms developed in the papers cited above are mainly concerned with regular cases (i.e., when boundary layers are absent). Fitted difference scheme on an equidistant mesh for the numerical solution of the linear threepoint reactiondiffusion problem have been studied in [15]. A uniform finite difference method, which is firstorder convergent, on an Smesh (Shishkin type mesh) for a singularly perturbed semilinear onedimensional convectiondiffusion threepoint boundary value problem have also been studied in [16].
Computational methods for singularly perturbed problems with two small parameters have been studied in different ways [17–21]. In this paper, we propose the hybrid scheme for solving the nonlocal problem (1.1)(1.2), which comprises three kinds of schemes, such as Samarskii's scheme [22], a finite difference scheme with uniform mesh, and finite difference scheme on piecewise uniform mesh. The considered algorithm is monotone.
We will prove that the method for the numerical solution of the threepoint boundary value problem (1.1)(1.2) is uniformly convergent of order on special piecewise equidistant mesh, in discrete maximum norm, independently of singular perturbation parameter. In Section 2, we present some analytical results of the threepoint boundary value problem (1.1)(1.2). In Section 3, we describe some monotone finitedifference discretization and introduce the piecewise uniform grid. In Section 4, we analyze the convergence properties of the scheme. Finally, numerical examples are presented in Section 5.
Notation 1.
Henceforth, denote the generic positive constants independent of and of the mesh parameter. Such a subscripted constant is also independent of and mesh parameter, but whose value is fixed.
Assumption.
In what follows, we will assume that , which is nonrestrictive in practice.
2. Properties of the Exact Solution
For constructing layeradapted meshes correctly, we need to know the asymptotic behavior of the exact solution. This behavior will be used later in the analysis of the uniform convergence of the finite difference approximations defined in Section 3. For any continuous function , we use for the continuous maximum norm on the corresponding interval.
Lemma 2.1.
If , and , the solution of (1.1)(1.2) satisfies the following estimates:
provided that and where
Proof.
3. Discretization and Piecewise Uniform Mesh
Introduce an arbitrary nonuniform mesh on the segment
Let be a mesh size at the node and be an average mesh size. Before describing our numerical method, we introduce some notation for the mesh functions. Define the following finite differences for any mesh function given on by
For equidistant subintervals of the mesh, we use the finite differences in the form
To approximate the solution of (1.1)(1.2), we employ a finite difference scheme defined on a piecewise uniform Shishkin mesh. This mesh is defined as follows.
We divide each of the intervals and into equidistant subintervals, and we divide into equidistant subintervals, where is a positive integer divisible by 4. The transition points and , which separate the fine and coarse portions of the mesh, are obtained by taking
where and are given in Lemma 2.1. In practice, we usually have , and so the mesh is fine on , and coarse on . Hence, if we denote the step sizes in , , and by , and , respectively, we have
so that
On this mesh, we define the following finite difference schemes:
where
with the usual piecewise linear basis functions
It is now necessary to define an approximation for the second boundary condition of (1.2). Let be the mesh point nearest to . Then, using interpolating quadrature formula with respect to and , we can write
where
Substituting into (3.13), for the second boundary condition of (1.2), we obtain
Based on (3.7) and (3.15), we propose the following difference scheme for approximating (1.1)(1.2):
4. Uniform Error Estimates
Let , . Then, the error in the numerical solution satisfies
where
and is defined by (3.14).
Lemma 4.1.
Let be the solution to (4.1). Then, the estimate
holds.
Proof.
The proof is almost identical to that of [16, 23].
Lemma 4.2.
Under the above assumptions of Section 1 and Lemma 2.1, the following estimates hold for the error functions and :
Proof.
The argument now depends on whether or and In the first case
and the mesh is uniform with for all . Therefore, from (3.9), we have
The same estimate is obtained for and in a similar manner.
In the second case
and the mesh is piecewise uniform with the mesh spacing and in the subintervals and , respectively, and in the subinterval . We have the estimate in and and the estimate in . In the layer region , the estimate reduces to
Hence,
The same estimate is obtained in the layer region in a similar manner. We now have to estimate for . In this case, we are able to rewrite as follows:
Since
it follows that
Also, if we rewrite the mesh points in the form , evidently
The last two inequalities together, (4.10), give the bound
Finally, we estimate for the mesh points and . For the mesh point , reduces to
Since
it then follows that
The same estimate is obtained for in a similar manner. This estimate is valid when only one of the values of or is equal to . Next, we estimate the remainder term . Suppose that , and the second derivative of on this interval is bounded. From (3.14), we obtain
Combining Lemmas and gives us the following convergence result.
Theorem 4.3.
Let be the solution of (1) and be the solution of (29). Then,
5. Algorithm and Numerical Results
In this section, we present some numerical results which illustrate the present method.

(a)
The difference scheme (3.16)–(3.19) can be rewritten as
(5.1)
where
System (5.1) and (3.19) is solved by the following factorization procedure:
It is easy to verify that
Therefore, the described factorization algorithm is stable.

(b)
We apply the numerical method (3.16)–(3.19) to the following problem:
(5.5)
with
The exact solution of the problem is
where
This has the typical boundary layers at and . In the computations in this section, we take
The error of the scheme is measured in the discrete maximum norm. For any values of and , the maximum pointwise errors and the uniform are calculated using
where is the exact solution of (5.5) and is the numerical solution of the finite difference scheme (3.16)–(3.19).The convergence rates are
The corresponding uniform convergence rates are computed using the formula
References
Nayfeh AH: Introduction to Perturbation Techniques. John Wiley & Sons, New York, NY, USA; 1993.
O'Malley RE Jr.: Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences. Volume 89. Springer, New York, NY, USA; 1991:viii+225.
Doolan EP, Miller JJH, Schilders WHA: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, Ireland; 1980:xv+324.
Farrell PA, Hegarty AF, Miller JJH, O'Riordan E, Shishkin GI: Robust Computational Techniques for Boundary Layers, Applied Mathematics. Volume 16. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:xvi+254.
Roos HG, Stynes M, Tobiska L: Robust Numerical Methods for Singularly Perturbed Differential Equations, ConvectionDiffusionReaction and Flow Problems, Springer Series in Computational Mathematics. Volume 24. 2nd edition. Springer, Berlin, Germany; 2008:xiv+604.
Linß T, Stynes M: A hybrid difference scheme on a Shishkin mesh for linear convectiondiffusion problems. Applied Numerical Mathematics 1999,31(3):255270. 10.1016/S01689274(98)001366
Savin IA: On the rate of convergence, uniform with respect to a small parameter, of a difference scheme for an ordinary differential equation. Computational Mathematics and Mathematical Physics 1995,35(11):14171422.
Sun GF, Stynes M: A uniformly convergent method for a singularly perturbed semilinear reactiondiffusion problem with multiple solutions. Mathematics of Computation 1996,65(215):10851109. 10.1090/S0025571896007533
Cziegis R: The numerical of singularly perturbed nonlocal problem. Lietuvas Matematica Rink 1988, 28: 144152.
Čiegis R: On the difference schemes for problems with nonlocal boundary conditions. Informatica 1991,2(2):155170.
Nakhushev AM: Nonlocal boundary value problems with shift and their connection with loaded equations. Differential Equations 1985,21(1):92101.
Sapagovas MP, Chegis RYu: Numerical solution of some nonlocal problems. Litovskiĭ Matematicheskiĭ Sbornik 1987,27(2):348356.
Liu B: Positive solutions of secondorder threepoint boundary value problems with change of sign. Computers & Mathematics with Applications 2004,47(89):13511361. 10.1016/S08981221(04)901289
Ma R:Positive solutions for nonhomogeneous point boundary value problems. Computers & Mathematics with Applications 2004,47(45):689698. 10.1016/S08981221(04)900569
Amiraliyev GM, Çakir M: Numerical solution of the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Mechanics 2002,23(7):755764. 10.1007/BF02456971
Cakir M, Amiraliyev GM: Numerical solution of a singularly perturbed threepoint boundary value problem. International Journal of Computer Mathematics 2007,84(10):14651481. 10.1080/00207160701296462
Gracia JL, O'Riordan E, Pickett ML: A parameter robust second order numerical method for a singularly perturbed twoparameter problem. Applied Numerical Mathematics 2006,56(7):962980. 10.1016/j.apnum.2005.08.002
Linß T, Roos HG: Analysis of a finitedifference scheme for a singularly perturbed problem with two small parameters. Journal of Mathematical Analysis and Applications 2004,289(2):355366. 10.1016/j.jmaa.2003.08.017
Linß T: Layeradapted meshes for convectiondiffusion problems. Computer Methods in Applied Mechanics and Engineering 2003,192(910):10611105. 10.1016/S00457825(02)006308
Clavero C, Gracia JL, Lisbona F: High order methods on Shishkin meshes for singular perturbation problems of convectiondiffusion type. Numerical Algorithms 1999,22(1):7397. 10.1023/A:1019150606200
O'Malley RE Jr.: Twoparameter singular perturbation problems for secondorder equations. Journal of Mathematics and Mechanics 1967, 16: 11431164.
Samarskii AA: Theory of Difference Schemes. M. Nauka, Moscow, Russia; 1971.
Amiraliyev G, Çakir M: A uniformly convergent difference scheme for a singularly perturbed problem with convective term and zeroth order reduced equation. International Journal of Applied Mathematics 2000,2(12):14071419.
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Çakır, M. Uniform SecondOrder Difference Method for a Singularly Perturbed ThreePoint Boundary Value Problem. Adv Differ Equ 2010, 102484 (2010). https://doi.org/10.1155/2010/102484
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DOI: https://doi.org/10.1155/2010/102484
Keywords
 Finite Difference Scheme
 Mesh Point
 Mesh Function
 Liquid Crystal Material
 Shishkin Mesh