Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem

Abstract

We consider a singularly perturbed one-dimensional convection-diffusion three-point boundary value problem with zeroth-order reduced equation. The monotone operator is combined with the piecewise uniform Shishkin-type meshes. We show that the scheme is second-order 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 three-point boundary value problem:

(1.1)

(1.2)

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 Navier-Stokes equations of fluid flow at high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, shear in second-order 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.

Three-point boundary value problems have been studied extensively in the literature. For a discussion of existence and uniqueness results and for applications of three-point 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 three-point reaction-diffusion problem have been studied in [15]. A uniform finite difference method, which is first-order convergent, on an S-mesh (Shishkin type mesh) for a singularly perturbed semilinear one-dimensional convection-diffusion three-point 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 three-point 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 three-point boundary value problem (1.1)-(1.2). In Section 3, we describe some monotone finite-difference 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 layer-adapted 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:

(2.1)

provided that and where

(2.2)

Proof.

The proof is almost identical to that of [16, 23].

3. Discretization and Piecewise Uniform Mesh

Introduce an arbitrary nonuniform mesh on the segment

(3.1)

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

(3.2)

For equidistant subintervals of the mesh, we use the finite differences in the form

(3.3)

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

(3.4)

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

(3.5)

so that

(3.6)

On this mesh, we define the following finite difference schemes:

(3.7)

where

(3.8)

(3.9)

(3.10)

(3.11)

with the usual piecewise linear basis functions

(3.12)

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

(3.13)

where

(3.14)

Substituting into (3.13), for the second boundary condition of (1.2), we obtain

(3.15)

Based on (3.7) and (3.15), we propose the following difference scheme for approximating (1.1)-(1.2):

(3.16)

(3.17)

(3.18)

(3.19)

4. Uniform Error Estimates

Let , . Then, the error in the numerical solution satisfies

(4.1)

where

(4.2)

and is defined by (3.14).

Lemma 4.1.

Let be the solution to (4.1). Then, the estimate

(4.3)

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 :

(4.4)

Proof.

The argument now depends on whether or and In the first case

(4.5)

and the mesh is uniform with for all . Therefore, from (3.9), we have

(4.6)

The same estimate is obtained for and in a similar manner.

In the second case

(4.7)

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

(4.8)

Hence,

(4.9)

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:

(4.10)

Since

(4.11)

it follows that

(4.12)

Also, if we rewrite the mesh points in the form , evidently

(4.13)

The last two inequalities together, (4.10), give the bound

(4.14)

Finally, we estimate for the mesh points and . For the mesh point , reduces to

(4.15)

Since

(4.16)

it then follows that

(4.17)

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

(4.18)

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,

(4.19)

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

(5.2)

System (5.1) and (3.19) is solved by the following factorization procedure:

(5.3)

It is easy to verify that

(5.4)

Therefore, the described factorization algorithm is stable.

• (b) We apply the numerical method (3.16)–(3.19) to the following problem:

  

  (5.5)

with

(5.6)

The exact solution of the problem is

(5.7)

where

(5.8)

This has the typical boundary layers at and . In the computations in this section, we take

(5.9)

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

(5.10)

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

(5.11)

The corresponding -uniform convergence rates are computed using the formula

(5.12)

Table 1 Approximate errors and and the computed orders of convergence on the piecewise uniform mesh for various values of and .