International Journal of Computer Mathematics

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Oct 1, 2005 - variable-mesh TAGE iterative method for the numerical solution of two-point ... http://www.informaworld.com/terms-and-conditions-of-access.pdf .... Substituting approximations (9a)–(9f) into (8) and neglecting higher-order terms, we obtain ...... [2] Jain, M.K., Iyengar, S.R.K. and Subramanyam, G.S., 1984, ...
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A third-order-accurate variable-mesh TAGE iterative method for the numerical solution of two-point non-linear singular boundary value problems R. K. Mohanty a; N. Khosla a a Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India Online Publication Date: 01 October 2005 To cite this Article: Mohanty, R. K. and Khosla, N. (2005) 'A third-order-accurate variable-mesh TAGE iterative method for the numerical solution of two-point non-linear singular boundary value problems', International Journal of Computer Mathematics, 82:10, 1261 - 1273 To link to this article: DOI: 10.1080/00207160500113504 URL: http://dx.doi.org/10.1080/00207160500113504

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International Journal of Computer Mathematics Vol. 82, No. 10, October 2005, 1261–1273

A third-order-accurate variable-mesh TAGE iterative method for the numerical solution of two-point non-linear singular boundary value problems R. K. MOHANTY* and N. KHOSLA Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India (Received 7 September 2004; in final form 13 October 2004) We propose a third-order-accurate variable-mesh two-parameter alternating group explicit (TAGE) iteration method for the numerical solution of the two-point singular boundary value problem u +

α  α u − 2 u = f (r), r r

0 < r < 1, α = 1 and 2

subject to boundary conditions u(0) = A, u(1) = B, where A and B are finite constants. We also discuss a Newton–TAGE iteration method for the third-order numerical solution of a two-point non-linear boundary value problem. The proposed method is applicable to singular and non-singular problems and is suitable for use on parallel computers. The convergence analysis is briefly discussed. Computational results are provided to illustrate the proposed TAGE iterative methods. Keywords: Variable mesh; Third-order method; TAGE method; Newton–TAGE method; Singular equation; Convection diffusion equation; Burgers equation C.R. Categories: G.1.7; G.1.3

1.

Introduction

We consider the application of the numerical method to the non-linear two-point boundary value problem (1) −u + f (r, u, u ) = 0, 0 < r < 1 The boundary conditions are u(0) = A,

u(1) = B

(2)

where A and B are finite constants. Keller [1] has given the conditions under which (1) together with (2) has a unique solution. We assume that these conditions are satisfied in the problem that we are considering. For the numerical solution of the above boundary value problem, we discretize the solution region *Corresponding author. Email: [email protected]

International Journal of Computer Mathematics ISSN 0020-7160 print/ISSN 1029-0265 online © 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207160500113504

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R. K. Mohanty and N. Khosla

[0, 1] such that 0 = r0 < r1 < r2 < · · · < rN +1 = 1. The finite difference approximation to the equation (1) is obtained on [0, 1] which consists of three grid points rk , rk+1 and rk−1 , where rk − rk−1 = hk and rk+1 − rk = hk+1 [2]. The mesh ratio is σk = hk+1 / hk . When σk = 1, it reduces to the constant mesh case [3]. The standard five-point discretization of equation (1) is obtained using third-order variable-mesh approximations for u and u . This requires the use of fictitious points outside the region of solution. The variable-mesh method of thirdorder accuracy which we present here is based on only three grid points. This means that no fictitious points for incorporating the boundary conditions are required. Difficulties have been experienced in the past in the computation of the variable-mesh difference method of thirdorder accuracy for the numerical solution of the two-point singular boundary value problem. The solution usually deteriorates in the vicinity of the singularity. In this paper, we modify our method in such a way that the solutions retain their order and accuracy everywhere in the solution region even in the vicinity of the singularity. We also discuss the two-parameter alternating group explicit (TAGE) and Newton–TAGE iteration methods proposed by Evans [4, 5] and Mohanty et al. [6] to solve both linear and non-linear variable-mesh difference equations. Since these methods are explicit and are coupled compactly they are suitable for use on parallel computers.

2. The third-order difference method Let the exact solution value of u at the grid point rk be Uk = u(rk ). Let uk be the approximate value of Uk at the grid point rk . We define Pk = σk2 + σk − 1, Qk = (σk + 1)(σk2 + 3σk + 1), Rk = σk (1 + σk − σk2 ) and Sk = σk (σk + 1). Then our third-order method is as follows. Let [Uk+1 − (1 − σk2 )Uk − σk2 Uk−1 ] U¯ k = (hk Sk ) [(1 + 2σk )Uk+1 − (1 + σk )2 Uk + σk2 Uk−1 ]  = U¯ k+1 (hk Sk )

(3a) (3b)

[−Uk+1 + (1 + σk )2 Uk − σk (2 + σk )Uk−1 ]  U¯ k−1 = (hk Sk )  f¯k+1 = f (rk+1 , Uk+1 , U¯ k+1 )

(3d)

 f¯k−1 = f (rk−1 , Uk−1 , U¯ k−1 )

(3e)

U¯¯ k = U¯ k −

σk (σk2

+ σk + 1)hk ¯ (fk+1 − f¯k−1 ) 6Qk

f¯¯k = f (rk , Uk , U¯¯ k ).

(3c)

(3f) (3g)

Then, at each interior mesh point rk , k = 1(1)N, the differential equation (1) is discretized using the formula [2] Uk+1 − (1 + σk )Uk + σk Uk−1 =

h2k [Pk f¯k+1 + Qk f¯¯k + Rk f¯k−1 ] + Tk 12

where Tk = O(h5k ),

U0 = u0 = A

and

UN +1 = uN +1 = B.

(4)

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For convergence, the coefficients on the right-hand side of equation (4) must be positive, i.e. Pk > 0, Qk > 0 and Rk > 0. Since σk > 0, we have the conditions σk2 + σk − 1 > 0

(5a)

1 + σk − σk2 > 0.

(5b)

From (5a) and (5b), we obtain the condition √ √ 5−1 5+1 < σk < . 2 2

(6)

We now consider the application of the difference formula (4) to the linear singular equation u = D(r)u + E(r)u + f (r),

0 0 and ω2 > 0 are the acceleration parameters of the TAGE method and u(s+1/2) is an intermediate vector. Because of the special block form of G1 and G2 , the inverses of (G1 + ω1 I) and (G2 + ω2 I) can be given explicitly. Therefore the TAGE iterative formula (15) can be given in explicit form, making it suitable for parallel computation. Combining (15a) and (15b), we have the TAGE iterative method u(s+1) = Gu(s) + g

(16)

where G = (G2 + ω2 I)−1 (G1 − ω2 I)(G1 + ω1 I)−1 (G2 − ω1 I) g = (G2 + ω2 I)−1 [I − (G1 − ω2 I)(G1 + ω1 I)−1 ]RH. It is clear that the TAGE iterative method (15) or (16) converges to the exact solution u = A−1 RH iff the spectral radius S(G) of the iteration matrix G is 0 and ω2 > 0. Let D be a diagonal matrix given by D = diag(1, d2 , d3 , . . . , dN ) where

 dM =

This implies

c1 c2 · · · cM−1 , a2 a3 · · · aM

M = 2(1)N,

(17)

cN aN +1 > 0.

 1 1 1 D−1 = diag 1, , , . . . , d2 d3 dN

where 1 = dM



a2 a3 · · · aM , c1 c2 · · · cM−1

(18)

M = 2(1)N.

Consider the matrices G∗1 and G∗2 which are similar to G1 and G2 , respectively, and are given by G∗1 = DG1 D−1

(19a)

G∗2 = DG2 D−1

(19b)

With the aid of (12), (17) and (18), we obtain from (19)   0 b1   b 2 γ2     γ 2 b3   ∗ G1 =   ..   .    bN −1 γN −1  0 γN −1 bN   0 b1 γ1  γ1 b2     ..   ∗ . G2 =     b γ N −2 N −2     γN −2 bN −1 0 bN

(20a)

(20b)

We note that G∗1 and G∗2 are real symmetric matrices for real a, b, c and γ . Let G∗ = (G2 + ω2 I)G(G2 + ω2 I)−1 = (G1 − ω2 I)(G1 + ω1 I)−1 (G2 − ω1 I)(G2 + ω2 I)−1 .

(21)

Then G∗ is similar to G. Let G∗∗ be a similar matrix to G∗ , given by G∗∗ = DG∗ D−1 .

(22)

Then, with the aid of (17), (18) and (21), we obtain from (22) G∗∗ = (G∗1 − ω2 I)(G∗1 + ω1 I)−1 (G∗2 − ω1 I)(G∗2 + ω2 I)−1 .

(23)

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Thus S(G) = S(G∗ ) = S(G∗∗ ) ≤ G∗∗ 2 ≤ (G∗1 − ω2 I)(G∗1 + ω1 I)−1 2 · (G∗2 − ω1 I)(G∗2 + ω2 I)−1 2

(24)

Since G∗1 and G∗2 are real symmetric and (G∗1 − ω2 I) commutes with (G∗1 + ω1 I)−1 , we have (G∗1 − ω2 I)(G∗1 + ω1 I)−1 2 = S[(G∗1 − ω2 I)(G∗1 + ω1 I)−1 ]



λ − ω2

= max



a≤λ≤ ¯ b¯ λ + ω1

(25)

Clearly, if G∗1 is positive definite, ω1 > 0 and ω2 > 0, then (G∗1 − ω2 I)(G∗1 + ω1 I)−1 2 < 1

(26)

(G∗2 − ω1 I)(G∗2 + ω2 I)−1 2 < 1

(27)





λ − ω2 η − ω1