A SCHWARZ METHOD FOR FOURTH-ORDER SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEM WITH DISCONTINUOUS SOURCE TERM

Abstract

A singularly perturbed reaction-diffusion fourth-order ordinary differential equation(ODE) with discontinuous source term is considered. Due to the discontinuity, interior layers also exist. The considered problem is converted into a system of weakly coupled system of two second-order ODEs, one without parameter and another with parameter ε multiplying highest derivatives and suitable boundary conditions. In this paper a computational method for solving this system is presented. A zero-order asymptotic approximation expansion is applied in the second equation. Then, the resulting equation is solved by the numerical method which is constructed. This involves non-overlapping Schwarz method using Shishkin mesh. The computation shows quick convergence and results presented numerically support the theoretical results.

1. Introduction

Singular Perturbation Problems (SPPs) arises in several branches of engineering and applied mathematics, including fluid flow at high Reynolds numbers, heat and mass transfer at high Péclet numbers, chemical reaction, control theory, semi conductor devices, nuclear physics, etc. It is well-known fact that the solution of these problems have multi-scale character. That is, there are thin layers where the solution varies rapidly, while away from the layer(s) the solutions behaves regularly and varies slowly. So, the numerical treatment of SPPs gives major computational difficulties and in recent years a large number of special purpose methods have been proposed to provide accurate numerical solutions. For more detailed discussion on the analytical and numerical treatment of SPPs we may refer the reader to the books [1] - [5] and a very recent literature survey [6]. Miller et al. [8] considered a parameter-uniform Schwarz method for a singularly perturbed reaction-diffusion problem with an interior layer. Kopteva et al. [9] discussed an overlapping Schwarz method for a singularly perturbed semi-linear reaction-diffusion problem with multiple solutions. They have used Bakhvalov and Shishkin type meshes to obtain second-order convergence. Chandru and Shanthi applied a boundary value technique for singularly perturbed boundary value problem of reaction-diffusion type with discontinuous source term in [10]. The same type of problem evaluated using hybrid difference scheme to obtain second-order convergence [11]. A fitted mesh method for singularly perturbed Robin type boundary value problem with non-smooth data discussed in [12]. Recently, Chandru et al. evaluated a two parameter singularly perturbed problem with discontinuous source term using hybrid difference scheme [13]. But only in few papers, numerical methods for higher-order differential equations with smooth and non-smooth cases are developed. Some methods are available in the literature in order to obtain numerical solution to singularly perturbed fourth-order differential equations when f is smooth on Ω = (0, 1) [7], [14] - [17] and f is non-smooth on Ω [18]. Motivated by the works of [8,18], a fourth-order singularly perturbed reaction-diffusion boundary value problem with discontinuous source term is considered:

where 0 < ε << 1 is a singular perturbation parameter. Define Ω− = (0, d), Ω+ = (d, 1), d ∈ Ω, to indicate the jump at d in any function [w](d) = w(d+)−w(d−), b(x) on (Ω− ∪ Ω+) and c(x) on = [0, 1] such that

Further it is assumed that f is sufficiently smooth on ╲{d}; a single discontinuity in the source term f(x) occurs at a point d ∈ Ω; f(x) and its derivatives have jump discontinuity at the same point. In general this discontinuity gives rise to interior layers in the second derivative of the exact solution of the problem. As f is discontinuous at d the solution y of (1)-(2) does not necessarily have a continuous fourth derivative at the point d. Thus y ∉ C4(Ω). However, the third derivative of the solution exists and is continuous.

This paper is organized as follows. Section 2 presents analytic behavior of the solution of the system of SPP (1)-(5). Some analytical and numerical results for second-order singularly perturbed boundary value problem with discontinuous source terms are described in Section 3.1, numerical scheme in Section 3.2 and truncation error analysis estimated in Section 3.3. The computational technique for the considered problem is discussed in Section 4. Section 5 explains the error estimates for the numerical solution. Numerical example is solved in Section 6. The paper ends with a conclusion. Throughout this paper, C denotes a generic positive constant that is independent of nodal point (i), number of mesh point (N) and the singular perturbation parameter ε. We use the norm ║w║ = supx∈Ω |w(x)|. Further means (|y1(x)|, |y2(x)|)T .

 

2. Some analytical results

In this section we derive a maximum principle for the following problem. Then using this principle, a stability result for the same problem is derived. Further, an asymptotic expansion approximation is constructed for the solution. Using the transformation y = y1 and , the SPBVP (1)-(2) can be transformed into an equivalent problem of the form

where = (y1, y2)T , y1 ∈ C2∩C3(Ω)∩C4(Ω− ∪Ω+), y2 ∈ C0∩C1(Ω)∩C2(Ω− ∪ Ω+) [18]. The proof of the Theorem 1-3 are obtained by following the steps defined in [18].

Theorem 1. The BVP (1)-(2) has a solution y ∈ C2∩C3(Ω)∩C4(Ω−∪Ω+).

Theorem 2 (Maximum principle). Suppose that = (y1, y2)T , y1 ∈ C2 ∩ C3(Ω) ∩ C4(Ω− ∪ Ω+), y2 ∈ C0 ∩ C1(Ω) ∩ C2(Ω− ∪ Ω+), satisfies and , ∀ x ∈ Ω− ∪ Ω+ and [y2]′(d) ≤ 0. Then , ∀ x ∈ .

Theorem 3 (Stability result). Consider the BVPs (6)-(7) subject to conditions (3)-(5). If y1 ∈ C2∩C3(Ω)∩C4(Ω−∪Ω+), y2 ∈ C0∩C1(Ω)∩C2(Ω−∪Ω+), then

2.1. Some asymptotic expansion approximation. Consider the BVP (6)-(7). Using the perturbation method defined in [2,15], we can construct an asymptotic expansion for the solution of the BVP (6)-(7) as follows. Let ūl0 = (ul01, ul02), and ūr0 = (ur01, ur02) be the solutions of the reduced problem (6)-(7).

subject to the conditions

and are layer correction terms given by

Here,

Now let be the right-layer corrections given by

where

The values of k1, k2, k3, k4 are determined by imposing the following boundary and continuity conditions:

Let

where u01 ∈ C2,

Remark 1. If (ul01, ur02) are the solution of (8)-(11), then u01 is the solution of the BVP

In the following it is assumed that BVP (12)-(13) can be solved exactly and closed form solution is available. This problem has a unique solution u01 ∈ C0 ∩ C1(Ω) ∩ C2(Ω− ∪ Ω+) [18].

Theorem 4. The zero-order asymptotic expansion approximation of the solution of (6)-(7) satisfies the inequality

where y1,as ∈ C2∩C3(Ω)∩C4(Ω−∪Ω+), y2,as ∈ C0∩C1(Ω)∩C2(Ω−∪Ω+).

Proof. It is easy to verify that y1,as ∈ C2 ∩ C3(Ω) ∩ C4(Ω− ∪ Ω+) and y2,as ∈ C0∩C1(Ω)∩C2(Ω−∪Ω+). Defining barrier functions as

it is easy to verify that

and

for a suitable selection of C. Then, by Theorem 2 we have the required result. □

 

3. Some analytical and numerical results for SPBVP for second-order ODEs with discontinuous source terms

We state some results for the following SPBVP which are needed in the rest of the paper. Consider the SPBVP

Remark 2. The BVP (14)-(15) has a unique solution ∈ C0 ∩ C1(Ω) ∩C2(Ω− ∪ Ω+) [19].

3.1. Analytical results.

Theorem 5. If (y1, y2) and are solutions of the BVPs (6)-(7) and (14)-(15), respectively, then

Proof. Since (y1, y2) are the solution of (6)-(7), then y2 satisfies the BVP

Further, the function w = y2 − satisfies the BVP

From Theorem 4 and the definition of v01 and w01, we have

that is,

From this inequality and the stability result given in [19] we have

that is,

The proof of the Theorem 6-7 are obtained by following the steps defined in [5].

Theorem 6. Let be the solution of (L) and be the corresponding sequences of Schwarz iterates. Then, for all k ≥ 1

where C is independent of k and ε and

The solution (x) of (L) is decomposed as

where u02(x) is the smooth component and v02(x) and w02(x) are the singular components. Each of the Schwarz iterates is also decomposed in an analogous manner. Thus, for k ≥ 1,

Theorem 7. Let u02 be the smooth component and v02, w02 are singular components of be the corresponding sequences of Schwarz iterates. Then, for all k ≥ 1 and for all x ∈ ,

where C is independent of k and ε and

3.2. Numerical scheme. A non-overlapping Schwarz iterative process for the BVP (14)-(15) is now described. On (Ω− ∪ Ω+) a piecewise uniform mesh of N mesh interval is constructed as follows. The interval Ω−; is subdivided into the three subintervals [0, τ1), [τ1, d−τ1) and [d−τ1, d) for some τ1 that satisfies 0 < τ1 ≤ d/4. On [0, τ1) and [d−τ1, d) a uniform mesh with N/8 mesh intervals is placed, while [τ1, d − τ1) has a uniform mesh with N/4 mesh intervals. The subinterval [d, d+τ2), [d+τ2, 1−τ2) and [1−τ2, 1] of Ω+ are treated analogously for some τ2 satisfying 0 < τ2 ≤ (1 − d)/4. The interior points of the mesh are denoted by

Clearly xN/2 = d and . Note that this mesh is a uniform mesh when τ1 = d/4 and τ2 = (1 − d)/4. It is fitted to the BVP (14)-(15) by choosing τ1 and τ2 to be the following functions of N and ε.

The discretization of (14)-(15) and the procedure for the non-overlapping Schwarz iterative technique is described below

Here the proper choice of an initial guess for the unknown . Then, we solve the following two finite difference subproblems for the mesh functions , k ≥ 1:

where,

After these two sub-problems are solved, the approximation to is updated using the average of the computed values at the two neighboring nodes of d. That is,

We define the kth approximation to as

where are the continuous linear interpolant of on Ω− and Ω+ respectively. The corresponding mesh functions are defined in the earlier stage of this section.

Each of the iterates is decomposed into a smooth component and singular components . Thus

where are defined, for all k ≥ 1,

3.3. Error analysis. The error at each mesh point is denoted by

Theorem 8. Let denote the smooth components of respectively. Then, for all k ≥ 1 and xi ∈

Proof. Using the result in [5](Lemma 1 pg.21),

and

Similarly for xi ∈ Ω+

Then it leads to the required estimate

Theorem 9. Let denote the singular components of respectively. Then, for all k ≥ 1

Proof. The argument lying on (0, τ1) and (d − τ1, d), the local truncation error of the singular part of the solution is estimated as follows

and for (τ1, d − τ1)

Using (24) outside the layers and at xi = τ1, xi = d − τ1 on Ω− gives

Using (25) outside the layers and at xi = τ1, xi = d − τ1 on Ω− gives

Hence

Then it leads to the required estimate

Similarly, the local truncation error of the singular component lying on (d + τ2) and (1 − τ2, 1) is

At the point xi = d.

Theorem 10. The error in using the scheme (18)-(21) to solve the BVP problem (14)-(15) at the inner grid points {xi, i = 1, 2, ..., N − 1} satisfies

Proof. From Theorem 8, 9 and (26) the above result can be obtained. □

 

4. Computational method

Consider the BVP (6)-(7). Let u01(x) be the reduced problem solution of the BVP (12)-(13) . From the Theorem 5 we get |y1(x) − u01(x)| ≤ . The first step in the computational method is to replace y1 by u01 in the second equation of the system (6)(as we have said earlier it is assumed that the closed from solution is available). Hence the system (6) gets decoupled. In the second step, we find the numerical approximation solution for y2 by applying the scheme (18)-(21). Then find y1 of (6) by using current y2 in the similar manner. This iterative process is repeated until successive iterates are sufficiently close at each point of , in the sense that they satisfy the converging criterion

 

5. Error estimate

Theorem 11. Let (y1, y2) be the solution of (6)-(7). Further, let be its numerical solution (14)-(15) obtained by numerical scheme. Then

Proof. Using Theorem 5 and 10 and the triangle inequality, we conclude that,

Remark 3. There are two boundary layers (x = 0 and x = 1) and an interior layer at x = d. If the boundary conditions happen to have values such that no boundary layer occurs at a boundary point and do the necessary modifications in the distribution of the mesh points[8].

Remark 4. So far, it has been assumed that the exact solution u01 of the BVP (12)-(13) is available. If not, one has to obtain a numerical solution for u01 by a suitable finite difference method with a piecewise uniform mesh of N mesh interval described in Section 3.2. As done earlier, in the second equation the values of y1 at the above grid points will be taken as u01,i, then the resulting equations are solved for y2,i.

 

6. Numerical results

In this section an example is solved for the particular problem of the type (1)-(2).

Example 1.

which validate the theoretical results established in the previous result. The maximum pointwise errors and number of iterations are evaluated using the double mesh principle.

where YN(xi) and Y2N(xi) denote the numerical solutions obtained using N and 2N mesh intervals. In addition, the order of convergence is calculated from

The solution is presented for various values of N and ε in Table 1.

TABLE 1.Maximum point-wise errors and iteration counts for various N and ε for the Problem 1.

 

7. Conclusion

A fourth-order singularly perturbed two point boundary value problem for ODEs with discontinuous source term is considered. The suitable boundary conditions are used to reduce the fourth-order differential equation into a system of two second-order equations and also established maximum principle, stability result and other necessary estimates. An iterative numerical method is used to solve the given example and numerical result is in agreement with the theoretical results.