Initial-Value Technique for Singularly-Perturbed ...

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 99, No. 1, pp. 37-52, OCTOBER 1998

Initial-Value Technique for Singularly-Perturbed Turning-Point Problems Exhibiting Twin Boundary Layers1,2 S. NATESAN3 AND M. RAMANUJAM 4

Communicated by F. E. Udwadia

Abstract. The initial-value technique that was originally developed for solving singularly-perturbed nonturning-point problems (Ref. 1) is used here to solve singularly-perturbed turning-point problems exhibiting twin boundary layers. In this method, the required approximate solution is obtained by combining solutions of the reduced problem, an initialvalue problem, and a terminal-value problem. Error estimates for approximate solutions are obtained. The implementation of the method on parallel architectures is discussed. Numerical examples are presented to illustrate the present technique. Key Words. Singularly perturbed boundary-value problems, asymptotic expansions, boundary layers, turning points, exponentially fitted difference schemes, initial-value techniques, parallel computation.

1. Introduction

Singular perturbation problems with or without turning point(s) are of common occurrence in many branches of applied mathematics such as fluid mechanics, elasticity, quantum mechanics, chemical reactor theory, etc. A few notable examples are boundary-layer problems, WKB problems, modeling of steady and unsteady viscous flow problems with large Reynolds number, and convective heat-transport problems with large Peclet number. In particular, the following singularly-perturbed turning-point problem (TPP) 1The

authors thank the referees for valuable comments. first author acknowledges the Council of Scientific and Industrial Research, New Delhi, India for financial support. 3Research Scholar, Department of Mathematics, Bharathidasan University, Tiruchirapalli, India. 4Professor, Department of Mathematics, Bharathidasan University, Tiruchirapalli, India. 2The

37 0022-3239/98/1000-0037$15.00/0 © 1998 Plenum Publishing Corporation

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JOTA: VOL. 99, NO. 1, OCTOBER 1998

received much attention in the literature because of the complexity involved in finding uniformly valid asymptotic expansions unlike nonturning-point problems:

where e is a small positive parameter and a, b,f are smooth functions such that a(x) has a simple zero at x = 0, that is,

This type of problem arises, e.g., as a linearized one-dimensional slice of a fluid flow problem having a region of recirculation. In Ref. 2, Abrahamsson derived a priori estimates for solutions of (1). O'Malley (Refs. 3-4), Roos et al. (Ref. 5), Wasow (Ref. 6), Watts (Ref. 7), Bender and Orszag (Ref. 8), Lagerstrom (Ref. 9), and Jindge (Ref. 10) have also studied qualitative aspects of these problems, namely existence, uniqueness, and asymptotic behavior of the solution. Farrell (Ref. 11) obtained a set of general sufficient conditions for a finite-difference scheme to be uniformly accurate for TPPs of the form (1) having a cusp-type boundary layer. Berger et al. (Ref. 12) modified the El-Mistikway and Werle scheme (Ref. 13) for TPPs. Kadalbajoo and Reddy (Ref. 14) presented a novel initial-value technique (IVT) for a class of nonlinear singularly-perturbed two-point BVPs for second-order ordinary differential equations (ODEs) without turning points. In their method, the original BVP is replaced by an asymptotically equivalent initial-value problem (IVP) for first-order ODEs, which is then solved by the Runge-Kutta method. In continuation of this work, Gasparo and Macconi (Ref. 15) used the IVT to solve a more general class of nonlinear singularly-perturbed BVPs without turning points. Further, Gasparo and Macconi (Ref. 1) used the IVT with a slight modification, and hence solved a class of linear and nonlinear singularly-perturbed BVPs without turning points. The asymptotic nature of the solutions of the singularly perturbed BVPs has been fully exploited by these authors in replacing the given two-point BVP with a terminal-value problem (TVP) and an IVP. The integration of these problems goes in opposite directions, but each problem can be solved independently of the other. In the present paper, we consider the BVP (1) subject to the following conditions:


39

Under these conditions, the BVP (1) has a unique solution exhibiting two boundary layers, one at x = — l and the other at x=l, and being smooth near x = 0 (Ref. 12). In Ref. 12, it has been proved that

where u is the solution of (1) and MO is the solution of the reduced problem

Further, it is mentioned in Ref. 12 that the number 1/2 in (3) is arbitrary. Also, since a(—1)>0 and a(l)0 and L 0 v(x)>0 for all xeD. Then,

Lemma 2.2. See Ref. 16. Consider the IVP (7). If u(x) is the solution of this IVP, then

To solve (7) numerically, we apply the exponentially fitted difference (EFD) schemes of Refs. 16-18 which are described below. (i)

Explicit EFD Scheme:


(ii)

41

Implicit EFD Scheme:

Remark 2.1. p i ( p ) is known as a fitting factor. The following lemma is the discrete analogue of Lemmas 2.1 and 2.2. Lemma 2.3. See Ref. 16. Let vi be a mesh function. Suppose that v0>0 and Lhpvi>0,_for all x i eD, p = 1 [ p = 2]. Then, vi>0 for all xieD. Moreover, for XjeD,

The following lemmas of Ref. 16 provide error estimates for the above EFD schemes. Lemma 2.4. See Ref. 16. Let u and ui be the solutions of (7) and (8). Then,

Lemma 2.5. See Ref. 16. Let u and ui be the solutions of (7) and (9). Then,

A well-known method of obtaining a higher-order approximation to the solution of an IVP for first-order ODEs is the Richardson extrapolation technique. Doolan et al. (Ref. 16) gave a necessary and sufficient condition for this technique to provide a higher-order approximation uniformly in e for the IVP (7). In fact, we have the following lemma. Lemma 2.6. See Ref. 16. Let u and uih be the solutions of (7) and (8). Then,

if and only if f/a is constant on D.

42


Motivated by the IVPs that arise in the IVT to be discussed in the next section, we consider the IVP

where a, b,f are smooth functions such that a ( x ) > a >0, xeD, and b(x) is bounded on D. Remark 2.2. From Lemma 2.2, it is obvious that the solution of (10) is bounded on D. Further, it may be noted that the IVP (10) is a perturbed problem of (7). The following theorems establish a relation between (7) and (10). Theorem 2.1. Let u and w be the solutions of (7) and (10). Then,

Proof.

Let

Then,

Lemma 2.2 yields the desired result. Consider the following explicit EFD scheme for (10):

where

Lemma 2.7. Let vi be a mesh function. Suppose that v 0 >0 and Lh3Vi>0 for all xieD. Then, vi>0 for all X i eD. Moreover, for XjeD,


Proof.

43

Since

the proof follows from Lemma 2.3. Remark 2.3. is bounded.

D

A consequence of Lemma 2.7 is that the solution of (11)

We have the following discrete analogue of Theorem 2.1, which follows from Lemma 2.7. Theorem 2.2.

Let ui and wi be the solutions of (8) and (11). Then,

Remark 2.4. It may be noted that the fitting factor p i ( p ) appearing in (11a) does not contain the coefficient b. Theorem 2.3.

Proof.

Let w and wi be the solutions of (10) and (11). Then,

By Theorem 2.1, Lemma 2.4, and Theorem 2.2, we have

Similarly, we have the following theorem. Theorem 2.4.

Consider the following implicit EFD scheme for (10):

where

Let w and wi be the solutions of (10) and (12). Then,

44


2.2. Initial-Value Technique. See Ref. 1. Consider the following singularly-perturbed second-order BVP:

where a, b, f are sufficiently smooth functions such that

Under these assumptions, the BVP (13) has a unique solution exhibiting a boundary layer of width O(e) at x = c. In order to get a numerical solution of (13) by the IVT, we proceed as follows (Ref. 1). Consider the following problems:

and

Then, we have

In this way, it is possible to obtain via (16) an 0(e)-approximation to u(x) by solving problems (14) and (15). These problems can be solved independently. Since the right-hand side of the ODE (15a) is zero, the IVP (15) can be solved by applying the Richardson extrapolation technique to yield a higher-order approximation (Lemma 2.6).

3. Error Estimates

Consider the BVP (13). We now state the maximum principle and uniform stability result for this BVP. Lemma 3.1. See Ref. 16. Let y be the smooth function satisfying Bcy(c)>0, Bdy(d)>0, and Ly(x)0, compute v02i by integrating (25) on [1/2, 1]. Step 4. For i>0, compute ui by

It is obvious that the parallelism is inherent in the above algorithm. Hence, separate processors can be assigned for Step 2 and Step 3, which lead to the following two-processor scheme. Parallel Scheme.

Task 1. Perform Step 2 on Processor P1. Task 2. Perform Step 3 on Processor P2. Task 3. Perform Step 4 by using the results of previous tasks. Remark 4.1. For a discussion of the reduced problem (4), one may refer to Pages 54, 59, 68 of Ref. 2 and to Refs. 3 and 5.

5. Numerical Examples

In this section, we present some numerical examples to illustrate the present method. For convenience, we take the interval [0,1] as the domain of definition of the ODE instead of [-1,1], and consequently x= 1/2 will be the turning point. The numerical computations have been carried out on the DEC Micro Vax II Computer system at Bharathidasan University, Tiruchirapalli, India. Example 5.1. Consider the following singularly-perturbed TPP (Ref. 9, p. 67):


49

The exact solution is given by

Here, u0(x) = 0; v01 and v02 are respectively the solutions of the following problems

and

The numerical results are presented in Table 1. Example 5.2. Consider the following TPP:

whose asymptotic expansion solution is given by

Table 2 presents computational results of the problem. Remark 5.1. The IVP (20) and the TVP (25) are solved here by the implicit EFD scheme (12). One may also apply the explicit EFD scheme (11) or the Richardson extrapolation procedure for the explicit EFD scheme (11). Our experiences with the above Examples 5.1 and 5.2 show that these schemes yield almost the same results.

6. Discussion In the literature, the initial-value technique has been applied to solve various singularly-perturbed BVPs for second-order ODEs without turning points as reported in Section 1 of this paper. Although we have not derived a similar technique for turning-point problems, we have used the IVT (originally developed for nonturning-point problems) to solve the present turningpoint problem, by suitably dividing the interval [-1,1] into three disjoint subintervals, namely [-1, -1/2], (-1/2,1/2), [1/2, 1]. That is, we have applied the IVT on the intervals [-1, -1/2] and [1/2,1]. To solve the IVP


50

Table 1. Numerical results for Example 5.1. Error

Numerical solution e = H=0.0001

0.0010 0.0020 0.1000 0.2000 0.2500

1.000000 0.818912 0.670644 0.549242 0.449835 0.368434 0.301776 0.247187 0.202481 0.165867 0.135880 0.000048 0.000000 0.000000 0.000000 0.000000

0.0E + 00 1.3E-04 2.3E-04 2.7E-04 2.9E-04 3.0E-04 2.9E-04 2.8E-04 2.6E-04 2.4E-04 2.2E-04 2.6E-07 5.3E-12 0.0E + 00 0.0E + 00 0.0E + 00

0.3000 0.4000 0.5000 0.6000 0.7000

0.000000 0.000000 0.000000 0.000000 0.000000

0.7500 0.8000 0.9000 0.9980 0.9990 1-10e

0.000000 0.000000 0.000000 0.000000 0.000048 0.135880 0.165867 0.202481 0.247187 0.301776 0.368434 0.449835 0.549242 0.670644 0.818912 1.000000

Nodal points 0.000

1e 2e

3e 4e 5e 6e 7e

8e 9c l0e

1-9e -8e

1-7e

1-6e 1-5e 1-4e 1-3e l-2e 1-le 1.000

Numerical solution e = H=-0.001

Error

Numerical solution 6 = H=0.00001

Error

1.000000 0.818749 0.670352 0.548855 0.449380 0.367935 0.301252 0.246656 0.201955 0.165356 0.135390 0.000046 0.000000 0.000000 0.000000 0.000000

0.0E + 00 1.3E-05 2.2E-05 2.7E-05 2.9E-05 3.0E-05 2.9E-05 2.8E-05 2.6E-05 2.4E-05 2.2E-05 3.4E-12 4.8E-16 0.0E + 00 0.0E + 00 0.0E + 00

1.000000 0.818733 0.670323 0.548816 0.449334 0.367885 0.301200 0.246603 0.201902 0.165305 0.135341 0.000045 0.000000 0.000000 0.000000 0.000000

0.0E + 00 1.3E-06 2.2E-06 2.7E-06 2.9E-06 3.0E-06 2.9E-06 2.8E-06 2.6E-06 2.4E-06 2.2E-06 3.3E-13 4.2E-17 0.0E+00 0.0E+00 0.0E+ 00

0.0E+00 0.0E+00 0.0E + 00 0.0E + 00 0.0E+00

0.000000 0.000000 0.000000 0.000000 0.000000

0.0E + 00 0.0E + 00 0.0E + 00 0.0E + 00 0.0E + 00

0.000000 0.000000 0.000000 0.000000 0.000000

0.0E+00 0.0E+00 0.0E+00 0.0E + 00 0.0E+ 00

0.0E+00 0.0E+00 0.0E + 00 5.3E-12 2.6E-07 2.2E-04 2.4E-04 2.6E-04 2.8E-04 2.9E-04 3.0E-04 2.9E-04 2.7E-04 2.3E-04 1.3E-04 0.0E+00

0.000000 0.000000 0.000000 0.000000 0.000046 0.135390 0.165356 0.201955 0.246656 0.301252 0.367935 0.449380 0.548855 0.670352 0.818749 1.000000

0.0E + 00 0.0E+00 0.0E+00 4.8E-16 3.4E-12 2.2E-05 2.4E-05 2.6E-05 2.8E-05 2.9E-05 3.0E-05 2.9E-05 2.7E-05 2.2E-05 1.3E-05 0.0E + 00

0.000000 0.000000 0.000000 0.000000 0.000045 0.135341 0.165305 0.201902 0.246603 0.301200 0.367885 0.449334 0.548816 0.670323 0.818733 1.000000

0.0E+00 0.0E+00 0.0E+00 4.2E-17 3.3E-13 2.2E-06 2.4E-06 2.6E-06 2.8E-06 2.9E-06 3.0E-06 2.9E-06 2.7E-06 2.2E-06 1.3E-06 0.0E+00

(20) and the TVP (25) that arise in this technique, we have used the EFD schemes, because they seem to be more appropriate to these problems as they are straightforward, provide good results, and take less time. The proposed method is easy to implement as one has to solve only an IVP and a TVP for first-order ODEs. Also, from the numerical results, the present method seems to be efficient and accurate. This method is suitable for parallel architectures, and hence the computation time is further reduced. It may be noted that we have also provided necessary error estimates for the present method.

51


Table 2. Initial-value technique profiles for Example 5.2. Nodal points

Numerical solution e=H=0.001

0.000

Error

Numerical solution e = H-0.0001

Error

Numerical solution e=H =0.00001

Error

0.0E + 00 4.8E-06 1.1E-05 1.6E-05 2.1E-05 2.6E-05 2.9E-05 3.1E-05 3.2E-05 3.3E-05 3.2E-05 4.2E-11 4.8E-21 0.0E+00 0.0E+00 0.0E + 00

1.000000 0.818729 0.670317 0.548807 0.449323 0.367872 0.301185 0.246586 0.201884 0.165281 0.135319 -0.000200 -0.040000 -0.200000 -0.400000 -0.500000

0.0E + 00 4.8E-07 1.1E-06 1.6E-06 2.1E-06 2.6E-06 2.9E-06 3.1E-06 3.2E-06 3.3E-06 3.2E-06 4.2E-12 0.0E+00 0.0E + 00 0.0E + 00 0.0E + 00

0.0010 0.0020 0.1000 0.2000 0.2500

1.000000 0.818579 0.670025 0.548374 0.448742 0.367134 0.300281 0.245505 0.200617 0.163824 0.133658 -0.02000 -0.040000 -0.200000 -0.400000 -0.500000

0.0E+00 0.0E + 00 0.0E+00

1.000000 0.818716 0.670291 0.548768 0.449270 0.367805 0.301103 0.246488 0.201769 0.165151 0.135168 -0.002000 -0.040000 -0.200000 -0.400000 -0.500000

0.3000 0.4000 0.5000 0.6000 0.7000

-0.600000 -0.800000 -1.000000 -1.200000 -1.400000

0.0E+00 0.0E+00 0.0E+00 0.0E + 00 0.0E + 00

-0.600000 -0.800000 -1.000000 -1.200000 -1.400000

0.0E+ 00 0.0E + 00 0.0E + 00 0.0E+00 0.0E+00

-0.600000 -0.800000 -1.000000 - .200000 - .400000

0.0E+00 0.0E+00 0.0E+00 0.0E+00 0.0E + 00

0.7500 0.8000 0.9000 0.9980 0.9990 1-10e l-9e l-8e 1-7e 1-6e 1-5e 1-4e 1-3e 1-2e 1-1e .0000

-1.500000 -1.600000 -1.800000 -1.941053 -1.591994 -1.591994 -1.502303 -1.392710 -1.258809 -1.095217 -0.895362 -0.651213 -0.352965 0.011360 0.456392 1.000000

0.0E+ 00 0.0E+00 0.0E+00 2.4E-17 3.3E-09 6.6E-04 6.9E-04 7.0E-04 7.1E-04 7.1E-04 6.8E-04 6.3E-04 5.5E-04 4.3E-04 2.5E-04 0.0E + 00

-1.500000 -1.600000 -1.800000 -1.997999 -1.996000 -1.593794 -1.503923 -1.394150 -1.260069 -1.096297 -0.896262 -0.651933 -0.353505 0.011000 0.456212 1.000000

0.0E + 00 0.0E + 00 0.0E + 00 1.2E-19 1.3E-11 6.9E-05 7.2E-05 7.4E-05 7.5E-05 7.4E-05 7.1E-05 6.6E-05 5.7E-05 4.4E-05 2.6E-05 0.0E+00

- .500000 - .600000 - .800000 - .997999 - .996000 - .593974 - .504085 - .394294 - .260195 - .096405 -0.896352 -0.652005 -0.353559 0.010964 0.456194 1.000000

0.0E + 00 0.0E+00 0.0E+00 1.2E-19 1.3E-I4 7.0E-06 7.2E-06 7.4E-06 7.5E-06 7.4E-06 7.2E-06 6.6E-06 5.8E-06 4.5E-06 2.6E-06 0.0E+00

\e 2f

3e 4f

5e 6e 7e

8e

9e l0e

0.0E + 00 4.8E-05

1.1E-04 1.6E-04 2.1E-04 2.5E-04 2.9E-04 3.1E-04 3.2E-04 3.3E-04 3.2E-04 4.7E-10 3.7E-18

References 1. GASPARO, M. G., and MACCONI, M., Initial-Value Methods for Second-Order Singularly Perturbed Boundary- Value Problems, Journal of Optimization Theory and Applications, Vol. 66, pp. 197-210, 1990. 2. ABRAHAMSSON, L. R., A Priori Estimates for Solutions of Singular Perturbations with a Turning Point, Studies in Applied Mathematics, Vol. 56, pp. 51-69, 1977.

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3. O'MALLEY, R. E., Introduction to Singular Perturbations, Academic Press, New York, New York, 1974. 4. O'MALLEY, R. E., Singular Perturbation Methods for Ordinary Differential Equations, Springer Verlag, New York, New York, 1991. 5. Roos, H. G., STYNES, M., and TOBISKA, L., Numerical Methods for Singularly Perturbed Differential Equations, Springer Verlag, New York, New York, 1996. 6. WASOW, W., Linear Turning Point Theory, Springer Verlag, New York, New York, 1985. 7. WATTS, A. M., A Singular Perturbation Problem with a Turning Point, Bulletin of Australian Mathematical Society, Vol. 5, pp. 61-73, 1971. 8. BENDER, C. M., and ORSZAG, S. A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, New York, 1978. 9. LAGERSTROM, P. A., Matched Asymptotic Expansions, Springer Verlag, New York, New York, 1988. 10. JINDGE, D., Singularly Perturbed Boundary-Value Problems for Linear Equations with Turning Points, Journal of Mathematical Analysis and Applications, Vol. 155, pp. 322-337, 1991. 11. FARRELL, P. A., Sufficient Conditions for the Uniform Convergence of a Difference Scheme for a Singularly Perturbed Turning Point Problem, SIAM Journal on Numerical Analysis, Vol. 25, pp. 618-643, 1988. 12. BERGER, A. E., HAN, H., and KELLOG, R. B., A Priori Estimates and Analysis of a Numerical Method for a Turning Point Problem, Mathematics of Computation, Vol. 4, pp. 465-492, 1984. 13. EL-MISTIKAWY, T. M., and WERLE, M. J., Numerical Method for Boundary Layers with Blowing: The Exponential Box Scheme, AIAA Journal, Vol. 16, pp. 749-751, 1978. 14. KADALBAJOO, M. K., and REDDY, Y. N., Initial-Value Technique for a Class of Nonlinear Singular Perturbation Problems, Journal of Optimization Theory and Applications, Vol. 53, pp. 395-406, 1987. 15. GASPARO, M. G., and MACCONI, M., New Initial-Value Method for Singularly Perturbed Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 63, pp. 213-224, 1989. 16. DOOLAN, E. P., MILLER, J. J. H., and SCHILDERS, W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, Ireland, 1980. 17. FARRELL, P. A., Uniform and Optimal Schemes for Stiff Initial-Value Problems, Computers and Mathematics with Applications, Vol. 13, pp. 925-936, 1987. 18. MILLER, J. J. H., Optimal Uniform Difference Schemes for Linear Initial-Value Problems, Computer and Mathematics with Applications, Vol. 12B, pp. 12091215, 1986.