Higher-Order Accurate Schemes for a Constant Coefficient Singularly Perturbed Reaction-Diffusion Problem of Boundary Layer Type

Transcription

1 Applied Mathematical Sciences, Vol., 6, no. 4, HIKARI Ltd, Higher-Order Accurate Schemes for a Constant Coefficient Singularly Perturbed Reaction-Diffusion Problem of Boundary Layer Type Brian J. M c Cartin Applied Mathematics, Kettering University 7 University Avenue, Flint, MI USA Copyright c 6 Brian J. M c Cartin. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper addresses the numerical approximation of linear, steady, one-dimensional, constant coefficient singular perturbation problems of reaction-diffusion type which exhibit boundary layers. Since the diffusion and reaction terms are of opposite sign, this is a special case of the linear source problem []. Herein, an original derivation of the hyperbolically fitted discretization [8], second-order accurate uniformly in the small parameter, due to Miller [7] is presented. It is then shown how to apply the Mehrstellenverfahren technique of L. Collatz [] to produce formal fourth-order accuracy. This is then the springboard to schemes of uniform fourth-order and formal sixth-order accuracy. Mathematics Subject Classification: 65L, 65L, 65L Keywords: singular perturbation, reaction-diffusion, boundary layer Introduction A singular perturbation problem is said to be of reaction-diffusion type if the order of the problem is reduced by two when the small parameter is set Deceased January 9, 6

2 64 Brian J. McCartin to zero []. This is to be distinguished from a convection-diffusion problem [9] where the corresponding reduction in order is only one. Depending on the relative signs of the diffusion and reaction terms, the solution to such a reaction-diffusion equation may exhibit either boundary layer behavior (the subject of the present paper) or become highly oscillatory [6]. While the most general of such problems are nonlinear, transient and multidimensional, this paper focuses on the linear, steady-state, one-dimensional version with constant coefficients. The variable coefficient problem is the subject of [5]. Within these confines, we present new methods of numerical approximation that are uniformly fourth-order accurate and formally sixth-order accurate. This will be achieved by coupling the method of exponential fitting [3] with the Mehrstellenverfahren technique of L. Collatz []. In the development that follows, the focus will be on the boundary layer case. We begin by examining the local truncation error of central differencing as applied to this problem as well as that of Miller s scheme [7]. It is then shown how to apply the Mehrstellenverfahren technique (Hermitian method) of L. Collatz [] to produce formal fourth-order accuracy. These considerations reveal that the respective local truncation errors predict the known asymptotic behaviors of these varied methods. Motivated by this observation, we next provide a new derivation of Miller s scheme for the case of constant coefficients. The approach taken herein amounts to the annihilation of all objectionable terms (i.e. terms that misbehave in the boundary layer) in the local truncation error. Moreover, this approach indicates how to achieve uniform fourth-order accuracy in the small parameter ɛ for constant coefficients by combining exponential fitting with Mehrstellenverfahren. This then provides the springboard to a scheme of formal sixth-order accuracy. Finally, we turn to a comparison of central differencing, Miller s scheme, Mehrstellenverfahren as well as our new schemes on a constant coefficient boundary layer problem with known exact solution. These results are indeed very encouraging. Reaction-Diffusion Equation The subject of our investigation is the steady, one-dimensional, reactiondiffusion equation with constant coefficients. The corresponding self-adjoint two-point boundary value problem may be cast in the form ɛ u (x) + c u(x) = f(x); u() = u L, u() = u R, () where we assume that < ɛ and c >. Physically, ɛ is the diffusion coefficient while c is the reaction coefficient. Since ɛ is a small parameter, we

3 Constant coefficient reaction-diffusion problem of boundary layer type 65 have here a singular perturbation problem. When (as herein) ɛ >, boundary layers may develop while, for ɛ <, the solution may be highly oscillatory [6]. In what follows, all functions appearing (especially, the source term f(x)) are assumed to possess sufficient smoothness to justify any operations to which we may subject them. Also, we define the parameter z := h c/ɛ, which measures the strength of reaction relative to diffusion when scaled by the mesh width h := /N of Figure. For the problems of primary interest in this study, we have z. Lastly, in the ensuing derivations, we will restrict ourselves to ɛ > (boundary layer case) where singular perturbation theory [8] reveals that the boundary layers may be as thin as O( ɛ) and that in such a boundary layer u = O(), u = O(/ ɛ), and u = O(/ɛ). Figure : Computational Grid 3 The Central Difference Scheme If we discretize the two-point boundary value problem, Equation (), using central differences, we obtain ɛ ui+ u i + u i h + c u i = f i. () The local truncation error of this scheme, which is easily established using Taylor series, is given by L.T.E. = h [ f i }{{} O() c i u i }{{} O(/ɛ) ] + O(h 4 ). (3)

4 66 Brian J. McCartin In order to numerically resolve any boundary layers that arise, we require that h ɛ (boundary layer width). Perusal of the above local truncation error then reveals that it is effectively O() due to the presence of the term containing u i. Thus, in spite of the fact that this scheme is formally second-order accurate, it is not uniformly accurate to any degree in the small parameter ɛ. 4 The Miller Scheme Miller [7] has developed a scheme for Equation () which is second-order uniformly accurate for constant c. This scheme is defined as follows: ɛ z [cosh z ] ui+ u i + u i h + c u i = f i. (4) The appearance of hyperbolic functions in this context is referred to as exponential fitting [8]. The local truncation error of this scheme is given by L.T.E. = h [ f i ] + O(h 4 ). (5) }{{} Notice that, in comparison to Equation (3), the undesirable term involving u i has been eliminated. Thus, the local truncation error is O(h ), independently of ɛ. The Miller scheme is thereby seen to be second-order accurate, uniformly in ɛ. O() 5 Mehrstellenverfahren By incorporating the principal term of the local truncation error given by Equation (5) into the Miller scheme, Equation (4), we obtain the formally fourth-order accurate scheme: ɛ z [cosh z ] ui+ u i + u i + c u h i = f i + h f i. (6) The inclusion of higher-order information concerning the source term f(x) is the hallmark of the Mehrstellenverfahren technique (Hermitian method) of L. Collatz []. The local truncation error of this scheme is given by L.T.E. = h c [ 4 ɛ u i }{{} O(/ɛ) + c 6 uiv i ɛ }{{} O(/ɛ ) 5 uvi i ] + O(h 6 ). (7) }{{} O(/ɛ 3 )

5 Constant coefficient reaction-diffusion problem of boundary layer type 67 Observe that, for fixed ɛ, the principal term of the local truncation error is now O(h 4 ) so that this Mehrstellenverfahren scheme is formally fourth-order accurate. However, with h ɛ in order to resolve the boundary layer, this leading term is actually O(h 4 /ɛ ) = O(h ). Thus, this Mehrstellenverfahren scheme is not uniformly fourth-order accurate in the small parameter ɛ. If evaluation of f (x) is either difficult or impossible, then the further approximation f i f i+ f i + f i h (8) may be made without loss of formal order of accuracy. 6 An Original Derivation of the Miller Scheme Motivated by the above observed correlation between the behavior of the local truncation error in the boundary layer and the order of uniform accuracy of the corresponding scheme, we next present a new derivation of the Miller scheme, Equation (4), for the case of constant c. We begin by considering the second central difference ɛ ui+ u i + u i h = ɛ n= hn (n)! u(n) i. (9) Keeping only the first term in the above sum would produce the central difference approximation. Although this would be formally second-order accurate, the neglected terms in the series have components that display excessive growth in the boundary layer. Specifically, { [ ɛ h n u (n) = ɛ h n ( ) ( ) ] } c c n ɛ f (n ) + f (n 4) + + f + ( cɛ ɛ ɛ )n u ( ) c n ( ) c n = [h n f (n ) + + h n f ] + h n (ɛu ), () ɛ ɛ }{{}}{{} O(h ) O() where we have used Equation () recursively. With h ɛ, detailed examination of each of the terms on the right hand side of Equation () reveals that only the last one is O(), the remaining terms being O(h ) or higher. Hence, if we want to achieve uniform second-order accuracy then we need to include all such terms when deriving an approximation from Equation (9). Thus, we make the boundary layer approximation ( ) c n u (n) u ɛ ()

6 68 Brian J. McCartin which, when inserted into Equation (9), yields the approximation Solving for u i u i+ u i + u i h n= and summing the infinite series yields u i ( ) hn c n (n)! u i. () ɛ z [cosh z ] ui+ u i + u i h, (3) which is precisely Miller s approximation. In [5], this markedly new derivation of Miller s scheme will provide the insight needed to generalize it, in a straightforward fashion, to the general case of variable c. However, we now show how to capitalize on this new perspective to achieve several enhancements to Miller s scheme for the case of constant c. 7 Fitted Mehrstellenverfahren For the case of constant c, the above alternative derivation of Miller s scheme hints at how to achieve uniform fourth-order accuracy without expanding the difference stencil. Simply include the penultimate term of Equation () when making the boundary layer approximation, Equation (). Any attempt to eke additional accuracy from Equation () in this fashion would indeed require an expansion of the difference stencil beyond three points due to the need to approximate fi iv or higher derivatives of the source term. The resulting modified Miller scheme for constant c is given by z u i + u i ɛ +c u [cosh z ] ui+ h i = f i + c ɛ [ The local truncation error of this scheme is given by z [cosh z ] ] f i. (4) L.T.E. = h4 36 [ f i iv ] + O(h 6 ). (5) }{{} Thus, the local truncation error is O(h 4 ), independently of ɛ. The Fitted Mehrstellenverfahren scheme is thereby seen to be fourth-order accurate, uniformly in ɛ! 8 The New Scheme in Equation (4), the further approxi- In order to obviate the need for f i mation f i O() f i+ f i + f i h (6)

7 Constant coefficient reaction-diffusion problem of boundary layer type 69 is invoked without any loss of either formal or uniform accuracy. The resulting new scheme z u i + u i ɛ [cosh z ] ui+ h +c u i = f i + c ɛ [ z ] f i + f i fi+ [cosh z ] h. (7) will be referred to as the McCartin scheme in what follows. The local truncation error of this scheme is given by L.T.E. = h4 4 [ f i iv ] + O(h 6 ). (8) }{{} Thus, the local truncation error is once again O(h 4 ), independently of ɛ. The McCartin scheme is likewise also seen to be fourth-order accurate, uniformly in ɛ! O() 9 The Blended Scheme Comparison of the principal parts of the local truncation errors of the Fitted Mehrstellenverfahren scheme, Equation (5), and the McCartin scheme, Equation (8), leads to the surprising conclusion that an appropriate linear combination of these two schemes will yield a blended scheme which is formally sixth-order accurate. Specifically, 3 5 L.T.E. F itted Mehrstellenverfahren + 5 L.T.E. McCartin = O(h 6 ). (9) The resulting new scheme z ɛ [cosh z ] u i+ u i + u i h +cu i = f i + ɛ c [ z ] 3h f i + (f i+ f i + f i ) [cosh z ] 5h. () will be referred to as the Blended McCartin scheme in what follows. Boundary Layer Test Problem We now present the results of some numerical experiments aimed at demonstrating the efficacy of the new schemes developed above. First, we present a model problem for which we know the exact solution. Then, this test problem is used to compare our schemes with those commonly used in the literature for the boundary layer case. These new schemes provide greatly improved results in all instances examined.

8 7 Brian J. McCartin ε =.3963 ε =.953 ε = ε =.4888 ε =.444 ε =.7 Figure : Birth of a Boundary Layer (solid=u(x; ɛ), dashed=u (x)) The constant coefficient two-point boundary value problem, ɛ u + c u = e αx sin (βx); u() = u L, u() = u R, () possesses the analytical solution determined by the method of undetermined coefficients c/ɛ x u(x; ɛ) = c e + c e c/ɛ x + A e αx sin (βx) + B e αx cos (βx), () where and A = c ɛ(α + β ) [c ɛ(α + β )] + [ɛαβ] ; B = ɛαβ [c ɛ(α + β )] + [ɛαβ], c = u R + (B u L ) e c/ɛ e α [A sin β + B cos β] sinh ( c/ɛ ) ; c = u L B c,

9 Constant coefficient reaction-diffusion problem of boundary layer type 7 while the reduced problem, corresponding to ɛ = and ignoring the boundary conditions, has solution u (x) = c eαx sin (βx). (3) Figure displays the development of the boundary layers as ɛ + for the case of c =, u L =, u R =, α =, β = π. The dashed curve denotes the solution to the reduced problem, Equation (3), while the solid curves correspond to the solution of the boundary value problem, Equation (), for ɛ = m (m = 8,..., 3). This test problem will now be used as the basis for comparison of the methods described above. In the maximum norm, we have [4, p. 57] u(x; ɛ) u h (x; ɛ) Ch p, (4) where h := /N is the mesh parameter, C is the asymptotic error constant and p is the asymptotic rate of convergence which is estimated as [4, p. 59] ( ) un/ u p p N := log. (5) u N u Figures 3 through 8 display, for each method, the exact (solid line) and numerical (circles) solutions on the left with the corresponding error (dots) on the right for ɛ = 3 and N = 3 where h = /N. Tables through 6 contain the corresponding maximum error and convergence rate estimated from Equation (5) for ɛ = m (m = 8,..., 3) and N = n (n = 5,..., 9). Table clearly shows the formal second-order accuracy of the Central Difference scheme as h for fixed ɛ. However, the nonuniform nature of this convergence is evidenced by the growth of the errors as ɛ for fixed h. This contrasts sharply with Table where the uniform second-order accuracy of the Miller scheme for h with fixed ɛ as well as ɛ with fixed h is prominently displayed. Likewise, the results of Table 3 clearly indicate the formal fourth-order accuracy of the Mehrstellenverfahren scheme while those of Table 4 testify to the uniform fourth-order accuracy of the Fitted Mehrstellenverfahren scheme. Table 5 reinforces the fact that the uniformity of the fourth-order convergence of the McCartin scheme is not sacrificed by making the approximation of Equation (6). Finally, the nonuniform sixth-order accuracy of the Blended McCartin scheme is in evidence in Table 6. The erratic results for N = 5 are almost certainly a numerical artifact due to finite-precision arithmetic which is unsurprising since they involve errors on the order of %!

10 7 Brian J. McCartin Conclusion In the preceding sections, new higher-order accurate schemes for the steady, one-dimensional, constant coefficient reaction-diffusion equation of boundary layer type have been presented. These included schemes of uniform fourthorder and formal sixth-order accuracy. Detailed derivations as well as extensive comparisons to existing schemes have been provided. The computational efficacy of the McCartin scheme is underscored by the fact that the maximum error for ɛ = 3 with N = is 3 6 which is less than half of the corresponding error of the Miller scheme with N = 5 thereby indicating that only N grid points need be used for comparable accuracy. These boundary layer test case results provide strong numerical evidence for the validity of the following: Conjecture (Equivalence of Uniform Consistency and Convergence) For f(x) sufficiently smooth and u(x) defined by L[u] := ɛ u (x) + c u(x) f(x) = ; u() = u L, u() = u R, k th -order uniform consistency is equivalent to k th -order uniform convergence. That is to say, if and only if where L h [u h ] = u u h < Ch k (C independent of ɛ) L h [u] = L[u] + C l h l, l=k lim C l = O(h k l ). h ɛ=h There are a number of directions in which this work could be extended. For example, the derivation for a non-uniform mesh would be straightforward but tedious. It is anticipated that this would result in the loss of one order of formal accuracy. Also, the fitted operators presented herein could be combined with corresponding fitted uniform meshes []. Finally, treatment of multiple spatial dimensions, transients, and nonlinearities would each be of substantial interest. For the extension of these methods to variable coefficient reaction-diffusion problems with boundary layers, see [5]. The extension to the treatment of the highly oscillatory case, where the diffusion and reaction terms are of the same sign, will appear in [6]. Acknowledgements. As always, the assistance of Barbara A. McCartin in the production of this manuscript is most gratefully acknowledged.

11 Constant coefficient reaction-diffusion problem of boundary layer type 73 References [] G. Birkhoff and S. Gulati, Optimal Few-Point Discretizations of Linear Source Problems, SIAM J. Numerical Analysis, (974), no. 4, [] L. Collatz, The Numerical Treatment of Differential Equations, Springer- Verlag, New York, NY, [3] E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 98. [4] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman & Hall/CRC, Boca Raton, Florida,. [5] B. J. McCartin, Higher-Order Accurate Schemes for a Variable Coefficient Singularly Perturbed Reaction-Diffusion Problem of Boundary Layer Type, Applied Mathematical Sciences, (6), no. 4, [6] B. J. McCartin, Higher-Order Accurate Schemes for a Constant Coefficient Singularly Perturbed Reaction-Diffusion Problem of Highly Oscillatory Type, Applied Mathematical Sciences, (6), to appear. [7] J. J. H. Miller, On the Convergence, Uniformly in ɛ, of Difference Schemes for a Two Point Boundary Singular Perturbation Problem, Chapter in Numerical Analysis of Singular Perturbation Problems, P. W. Hemker and J. J. H. Miller (Eds.), Academic Press, New York, NY, 979, [8] J. J. H. Miller, E. O Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, Revised Edition, World Scientific, Singapore,. [9] K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, 996. [] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singular Perturbation Problems, Springer-Verlag, New York, NY, 996. [] G. I. Shishkin and L. P. Shishkina, Difference Methods for Singular Perturbation Problems, Chapman & Hall/CRC, Boca Raton, Florida, 8.

12 74 Brian J. McCartin Central Difference Scheme.5 error Figure 3: Central Difference Scheme (ɛ = 3, N = 3) ɛ N = 3 N = 64 N = 8 N = 56 N = ( 3) 9.594( 4).3478( 4) 5.88( 5).45495( 5) ( 3).8663( 3) ( 4).84( 4).9533( 5) ( ) 3.736( 3) ( 4).37636( 4) ( 5) ( ) ( 3).88546( 3) ( 4).945( 4) ( ).4647( ) 3.749( 3) 9.5( 4).3879( 4) ( ).4833( ) ( 3).88993( 3) ( 4) Table : Central Difference Scheme: Error and Convergence Rate

13 Constant coefficient reaction-diffusion problem of boundary layer type 75 Miller Scheme 4 x 4 error Figure 4: Miller Scheme (ɛ = 3, N = 3) ɛ N = 3 N = 64 N = 8 N = 56 N = ( 3) ( 4).9537( 4).747( 5) 6.854( 6) ( 3) 4.59( 4).333( 4).8363( 5) 7.99( 6) ( 3) ( 4).4886( 4).8788( 5) 7.3( 6) ( 3) 4.546( 4).55( 4).8943( 5) 7.448( 6) ( 3) ( 4).485( 4).8976( 5) 7.697( 6) ( 3) 4.598( 4).3566( 4).8943( 5) ( 6) Table : Miller Scheme: Error and Convergence Rate

14 76 Brian J. McCartin Mehrstellenverfahren Scheme 6 x 4 error Figure 5: Mehrstellenverfahren Scheme (ɛ = 3, N = 3) ɛ N = 3 N = 64 N = 8 N = 56 N = ( 5).4368( 6) ( 8) ( 9) () ( 5).8697( 6).79578( 7).337( 8) 7.35() ( 5) ( 6) 3.678( 7).655( 8).465( 9) ( 4).46( 5) ( 7) ( 8).84( 9) ( 4).44( 5).4459( 6) 9.769( 8) ( 9) ( 4) ( 5).8563( 6).87( 7).365( 8) Table 3: Mehrstellenverfahren Scheme: Error and Convergence Rate

15 Constant coefficient reaction-diffusion problem of boundary layer type 77 Fitted Mehrstellenverfahren Scheme 5 x 7 error Figure 6: Fitted Mehrstellenverfahren Scheme (ɛ = 3, N = 3) ɛ N = 3 N = 64 N = 8 N = 56 N = ( 7) 3.948( 8).47486( 9).54789() () ( 7) ( 8).69646( 9).68793().6() ( 7) 4.56( 8).8489( 9).78558().93() ( 7) 4.594( 8).94( 9).84999().588() ( 7) ( 8).9855( 9).88887().845() ( 7) ( 8).9787( 9).98().4() Table 4: Fitted Mehrstellenverfahren Scheme: Error and Convergence Rate

16 78 Brian J. McCartin McCartin Scheme 4 x 7 error Figure 7: McCartin Scheme (ɛ = 3, N = 3) ɛ N = 3 N = 64 N = 8 N = 56 N = ( 7) 5.939( 8) ( 9).38().4538() ( 6) ( 8) ( 9).535().5775() ( 6) ( 8) 4.79( 9).67934().679() ( 6) ( 8) ( 9).7775().7365() ( 6) 6.999( 8) 4.57( 9).83783().77683() ( 7) ( 8) ( 9).874().8() Table 5: McCartin Scheme: Error and Convergence Rate

17 Constant coefficient reaction-diffusion problem of boundary layer type 79 Blended McCartin Scheme 3 x 8 error Figure 8: Blended McCartin Scheme (ɛ = 3, N = 3) ɛ N = 3 N = 64 N = 8 N = 56 N = ( 9) 3.756() 5.833(3).599(4).78746(4) ( 9) 8.6().83().38778(4) 9.484(4) ( 8).734().734() (4) 6.74(4) ( 8) () () (4).465(4) ( 8) ().546().888(3) (5) ( 8).35956( 9).3() (3) (5) Table 6: Blended McCartin Scheme: Error and Convergence Rate Received: January 3, 6; Published: March 3, 6