Application of TAGE Iterative Methods for the Solution of Nonlinear

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Application of TAGE Iterative Methods for the Solution of Nonlinear Two Point Boundary Value Problems with Linear Mixed Boundary Conditions on a Non-Uniform Mesh a

a

R. K. Mohanty , Jyoti Talwar & Noopur Khosla

b

a

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India b

Department of Mathematics, Bharati Vidyapeeth's College of Engineering, Guru Gobind Singh Indraprastha University, New Delhi, India Available online: 27 Apr 2012

To cite this article: R. K. Mohanty, Jyoti Talwar & Noopur Khosla (2012): Application of TAGE Iterative Methods for the Solution of Nonlinear Two Point Boundary Value Problems with Linear Mixed Boundary Conditions on a Non-Uniform Mesh, International Journal for Computational Methods in Engineering Science and Mechanics, 13:3, 129-134 To link to this article: http://dx.doi.org/10.1080/15502287.2012.660228

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International Journal for Computational Methods in Engineering Science and Mechanics, 13:129–134, 2012 c Taylor & Francis Group, LLC Copyright ISSN: 1550-2287 print / 1550-2295 online DOI: 10.1080/15502287.2012.660228

Application of TAGE Iterative Methods for the Solution of Nonlinear Two Point Boundary Value Problems with Linear Mixed Boundary Conditions on a Non-Uniform Mesh R. K. Mohanty,1 Jyoti Talwar,1 and Noopur Khosla2 1

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India Department of Mathematics, Bharati Vidyapeeth’s College of Engineering, Guru Gobind Singh Indraprastha University, New Delhi, India

Downloaded by [University of Delhi] at 23:46 29 April 2012

2

We report the application of two parameter alternating group explicit (TAGE) iteration and Newton-TAGE iteration methods for the solution of nonlinear differential equation u = F(x, u, u ) subject to linear mixed boundary conditions on a non-uniform mesh. In both cases, we use only three non-uniform grid points. The convergence theory for TAGE iteration method is analyzed. Numerical examples are considered to demonstrate computationally and the utility of TAGE iteration methods. Keywords

Non-uniform mesh, Linear mixed boundary conditions, TAGE method, Newton-TAGE method, SOR method, Burgers’ equation

1. INTRODUCTION Consider the second-order non-linear ordinary differential equation u = F (x, u, u ), a ≤ x ≤ b.

(1)

The mixed boundary conditions in linear form at both ends are given by α0 u(a) − α1 u (a) = A, β0 u(b) + β1 u (b) = B,

(2a) (2b)

where −∞ < a ≤ x ≤ b < ∞, A, B are finite constants and α0 , α1 , β0 , β1 ≥ 0, α0 + α1 > 0, β0 + β1 > 0, α0 + β0 > 0. We assume that for x ∈ [a, b] and −∞ < u, v < ∞

This research was supported by The Council of Scientific and Industrial Research under research grant No. 09/045(0836)2009-EMR-I. Address correspondence to R.K. Mohanty, Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India. E-mail: [email protected]

(i) F (x, u, v) is continuous; (ii) Fu and Fv exist and are continuous; (iii) Fu > 0 and |Fv | < W for some positive constant W. These conditions assure us that the boundary value problems (1) and (2) have a unique solution (see [1]). The best known uniform mesh fourth-order finite difference method for the above boundary value problem is that of Chawla [2]; this method involves three grid points, constant values of step-length h > 0 at each interior grid point and three evaluations of F. Jain et al. [3] have given a third-order variable mesh three–point finite difference discretization for u = F (x, u, u ) subject to natural boundary conditions; their method involves four evaluations of F. Recently, Evans and Mohanty [4] have proposed a three-point second-order finite difference discretization of order two on a non-uniform mesh for the numerical solution of u = F (x, u, u ) subject to natural boundary conditions and discussed the application of the alternating group explicit method; their method involves only two evaluations of F. Further, Mohanty [5] has also developed a new third-order variable mesh method for the solution of u = F (x, u, u ) subject to Dirichlet type boundary conditions; he derives his scheme using only three grid points and three evaluations of function F. In this paper, we give a three point, third-order variable mesh finite difference discretization for the nonlinear boundary value problem (1) with linear mixed boundary conditions (2) and discuss the applications of two parameter alternating group explicit (TAGE) and Newton–TAGE iteration methods (see [6]). An interesting feature of our method is that each discretization of the differential equation at an interior grid point is based on just three evaluations of F and at the boundary involves just two evaluations of F. In the next section, we describe the present method. In Section 3, we discuss the application and error analysis of the proposed TAGE method (see [7, 8]). In Section 4,

129

130

R. K. MOHANTY ET AL.

we provide numerical results to illustrate the methods. We also compare our TAGE and Newton-TAGE iteration methods with the corresponding SOR and Newton-SOR (see [9–12]) methods.

Next, let F0 = F (x0 , U0 , U0 ), U 1/2


2. THE THIRD ORDER VARIABLE MESH METHOD We discretize the solution region [a,b] such that a = x0 < x1 < ........... < xN < xN = b. The finite difference discretization to the differential Eq. (1) is obtained on [a,b] that consists of three grid points xk , xk+1 and xk−1 , where hk = xk − xk−1 and hk+1 = xk+1 − xk . Let σk = (hk+1 /hk ) > 0, k = 1(1)N-1 be the mesh ratio parameter, where N is a positive integer. When σk = 1, it reduces to the constant mesh case. Let Uk = u(xk ) be the exact solution value of u(x) and at the grid point xk and uk be its approximate value. At the grid point xk , we denote Pk = σk2 + σk − 1, Qk = (σk + 1) σk2 + 3σk + 1 , Rk = σk 1 + σk − σk2 , Sk = σk (σk + 1), −σk 1 + σk + σk2 . γk = 6Qk

(3a) (3b)

U k = U k + γk hk (F k+1 − F k−1 ),

(4f)

F k = F (xk , Uk , U k ).

(4g)

Then, at each internal mesh point xk , k = 1(1)N − 1, the differential Eq. (1) is discretized by Uk+1 − (1 + σk )U k + σk Uk−1 h2 = k [Pk F k+1 + Qk F k + Rk F k−1 ] + Tk , 12

(5)

where Tk = O(h5k ) (see [5]). For convergence, the coefficients associated with the righthand side of Eq. (5) must be positive from which we obtain the condition √ √ 5−1 5+1 < σk < (See Jain et al [3]). (6) 2 2

h21 [F0 + 2F 1/2 ] + T0 , 6

α0 U0 − α1 U0 = A.

(7d)

(8)

(9)

Also let FN = F (xN , UN , UN ),

(10a) h2N

hN U + FN , (10b) 2 N 8 1 3 hN (UN − UN−1 ) + UN − FN , (10c) = 4hN 4 8

U N−1/2

(4e)

(7c)

where T0 = O(h51 ) and from (2a)

(3e)

(4d)

U1 = U0 + h1 U0 +

U N−1/2 = UN −

F (xk−1 , Uk−1 , U k−1 ),

(7b)

Then, at x0 we use the discretization

(3d)

F k+1 = F (xk+1 , Uk+1 , U k+1 ), F k−1 =

F 1/2 = F (x1/2 , U 1/2 , U 1/2 ).

(3c)

Our third-order variable mesh method is defined as follows: For k = 1(1)N-1, let U k = Uk+1 − (1 − σk2 )U k − σk2 Uk−1 /(hk Sk ), (4a) U k+1 = (1 + 2σk )U k+1 − (1 + σk )2 Uk + σk2 Uk−1 /(hk Sk ), (4b) 2 U k−1 = −Uk+1 + (1 + σk ) Uk − σk (2 + σk )Uk−1 / (hk Sk ) , (4c)

U 1/2

(7a)

h1 h2 = U0 + U0 + 1 F0 , 2 8 1 3 h1 (U1 − U0 ) + U0 + F0 , = 4h1 4 8

F N−1/2 = F (xN−1/2 , U N−1/2 , U N−1/2 ).

(10d)

Then, at xN , we use the discretization UN−1 = UN − hN UN +

h2N FN + 2F N−1/2 + TN , 6

(11)

where TN = O(h5N ) and from Eq. (2b) β0 UN + β1 UN = B.

(12)

If α1 = 0, then U0 = A/α0 is to be substituted in the difference Eq. (5); or else, if α1 = 0, then U0 = (α0 U0 − A)/α1 is to be substituted in Eq. (8). Similarly, if β1 = 0, then UN = B/β0 is to be substituted in the Eq. (5); or else, if β1 = 0, then UN = (B − β0 UN )/β1 , is to be substituted in Eq. (11). Note that the proposed method is applicable to non-singular problems when α1 = 0, β1 = 0 and applicable to both singular and nonsingular problems when α1 = β1 = 0 (see [5]). For constant mesh case, i.e., when hK = hk+1 the proposed method reduces to the method given by Chawla [2]. If the differential equation is linear, the resulting tri-diagonal linear system can be solved by TAGE method; in the nonlinear case, the system can be solved using the Newton-TAGE method, which are suitable for use on parallel computers. 3. TAGE ALGORITHM AND ERROR ANALYSIS Now consider the test equation u = D(x)u + E(x)u,

a≤x≤b

(13)

131

TAGE METHOD FOR MIXED BOUNDARY VALUE PROBLEMS

By the help of method Eq. (8) and using relation Eq. (9), we may discretize the differential Eq. (13) at x0 as 2b0 u0 + c0 u1 = RH0 ,

(14)


D0 = D(x0 ), E0 = E(x0 ), D1/2 = D(x0 + h1 /2), E1/2 = E(x0 + h1 /2), h2 h2 h2 h2 h1 A1 = 1+ 1 E0 − D1/2 1− 1 E0 + 1 E1/2 1+ 1 E0 , 6 4 6 3 8 2 h h h1 h h1 A2 = 1+ 1 D0 + D1/2 1+ 1 D0 + 1 E1/2 1+ D0 , 6 12 2 6 4 α0 2b0 = A1 + h1 A2 , α1 = 0, α1 h1 c0 = −1+ D1/2 , 4 Ah1 A2 , α1 = 0. RH0 = α1 Now, with the help of method Eq. (5), we can discretize the differential Eq. (13) at xk as

Ek = Lk = Mk = ak =

2bk =

ck =

RHk =

EN = E(xN ) DN−1/2 = D(xN − hN /2),

h2 h2N hN EN + DN−1/2 1 − N EN 6 4 6 2 2 h h + N EN−1/2 1 + N EN , 3 8 h hN hN DN−1/2 1 − DN = 1 − N DN − 6 12 2 2 h hN + N EN−1/2 1 − DN , 6 4 hN DN−1/2 , = −1 − 4 β0 = B1 + hN B2 , β1 = 0, β1 BhN = B2 , β1 = 0. β1

B1 = 1 +

B2

aN 2bN RHN

Note that the difference methods Eqs. (14), (15), and (16) have local truncation errors of O(h5k ), k = 1(1)N and are applicable to non-singular problems only. Assume that α1 = 0, β1 = 0. Combining the formulas Eqs. (14)–(16), we obtain the matrix equation D(xk ), Dk+1 = D (xk + hk+1 ) , Dk−1 = D (xk −hk ) , E(xk ), Ek+1 = E (xk + hk+1 ) , Ek−1 = E (xk −hk ) , 1 Au = R H, (17) Pk − σk σk2 + σk + 1 Dk hk , 6 1 Rk + σk σk2 + σk + 1 Dk hk , 6 where hk 2 −σk Qk Dk + σk2 Lk Dk+1 −σk + 12Sk ⎡ ⎤ h2 2b0 c0 0 − σk (2 + σk )Mk Dk−1 + k Mk Ek−1 , ⎢ ⎥ 12 c1 ⎢ a1 2b1 ⎥ ⎢ ⎥ hk 2 2 ⎢ ⎥ .. − 1 − σk Qk Dk − (1 + σk ) Lk Dk+1 1 + σk + A = , ⎢ ⎥ . 12Sk ⎢ ⎥ ⎢ ⎥ 2 h aN−1 2bN−1 cN−1 ⎦ ⎣ + (1 + σk )2 Mk Dk−1 + k Qk Ek , 12 0 aN 2bN (N+1)×(N+1) hk ⎤ ⎤ ⎡ ⎡ Qk Dk + (1 + 2σk )Lk Dk+1 − Mk Dk−1 −1 + u0 RH0 12Sk ⎢ u1 ⎥ ⎢ RH1 ⎥ ⎥ ⎢ ⎥ ⎢ h2 ⎢ .. ⎥ ⎢ .. ⎥ + k Lk Ek+1 , ⎥ ⎢ ⎢ 12 u=⎢ . ⎥ , RH = ⎢ . ⎥ ⎥ ⎥ ⎢ ⎢ . ⎥ . 0. ⎣ .. ⎦ ⎣ .. ⎦

ak uk−1 + 2bk uk + ck uk+1 = RHk ,

Dk =

DN = D(xN ),

EN−1/2 = E(xN − hN /2),

where

where

where

k = 1(1)N − 1,

(15)

Finally, with the help of method Eq. (11) and using relation Eq. (12), we may discretize differential Eq. (13) at the boundary point xN as aN uN−1 + 2bN uN = RHN ,

(16)

uN

(N+1)×1

RHN

(N+1)×1

Now we consider the case of N odd; the case N even follows easily.

132


We split the matrix A into the sum of two matrices A = G1 + G2 , where ⎤ ⎡ b0 c0 0 ⎥ ⎢ ⎥ ⎢ a1 b1 ⎥ ⎢ ⎥ ⎢ b c 2 2 ⎥ ⎢ ⎥ ⎢ a3 b3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. G1 = ⎢ , ⎥ . ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b c N−1 N−1 ⎦ ⎣ 0 aN bN


⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ G2 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

where G = (G 2 + ρ2 I)−1 (G 1 −ρ2 I ) (G 1 + ρ1 I)−1 (G 2 −ρ1 I), (23) is the TAGE iterative matrix and b = (G 2 + ρ2 I)−1 [I − (G 1 − ρ2 I)(G 1 + ρ1 I)−1 ]R H is the right–hand side vector. Let ε (s) = U − u(s) be the error vector at s th - iterate, then subtracting Eqs. (18a), (18b) from Eqs. (21a), (21b), we obtain the error equation ε(s+1) = Gε (s) ,

(24)

(N+1)×(N+1)

⎤

0

b0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

b1 c1 a2 b2 .. . ..

. bN−2 cN−2 aN−1 bN−1

0

where G is given by Eq. (23). For convergence, it is required to show that S(G) 0 and ρ2 > 0, where S(G) denotes the spectral radius of G. The eigenvalues λ of G 1 are given by

bN

.

k = 0(2)N − 1,

(25)

where ck ak+1 > 0. The eigenvalues η of G 2 are given by η = b0 , bN and (N+1)×(N+1)

The TAGE iterative method to solve Eq. (17) may be written as (G 1 + ρ1 I) v (s) = R H − (G 2 − ρ1 I) u(s) , s ≥ 0, (G 2 + ρ2 I) u(s+1) = R H − (G 1 − ρ2 I) v (s) , s ≥ 0

(18a) (18b)

where (G 1 + ρ1 I) and (G 2 + ρ2 I) are non-singular and ρ1 > 0, ρ2 > 0 are the acceleration parameters of the TAGE method and v (s) is an intermediate vector. Because of (2 × 2) block form of G 1 and G 2 , (G 1 + ρ1 I)−1 and (G 2 + ρ2 I)−1 can be written explicitly and so the TAGE method the Eq. (18) can be expressed in explicit form, making it suitable for use on parallel computers. We may re-write Eq. (17) in the form (G 1 + G 2 )u = R H,

(19)

where the exact solution U satisfies the equation (G 1 + G 2 )U = R H.

(G 1 + ρ1 I) U = R H − (G 2 − ρ1 I) U, (G 2 + ρ2 I) U = R H − (G 1 − ρ2 I) U.

(η − bk )(η − bk+1 ) − ck ak+1 = 0,

k = 1(2)N − 2,

(26)

Let G ∗ be a similar matrix to G and given by G ∗ = (G 2 + ρ2 I) G (G 2 + ρ2 I)−1 = (G 1 − ρ2 I) (G 1 + ρ1 I)−1 (G 2 − ρ1 I) × (G 2 + ρ2 I)−1 .

(27)

Using the property of matrix norms, we have

G ∗ 2 ≤ (G 1 − ρ2 I )(G 1 + ρ1 I)−1 2 · (G 2 − ρ1 I)(G 2 + ρ2 I )−1 2 . Further, since Re(λk ) > 0 and Re(ηk ) > 0, hence

λk − ρ2 −1

< 1.

(G 1 − ρ2 I)(G 1 + ρ1 I) 2 = Max λK λk + ρ1

(28)

(29a)

Similarly, (20)

From Eq. (18), we have two equivalent forms

(G 2 − ρ1 I)(G 2 + ρ2 I)−1 2 < 1.

(29b)

S(G) = S(G ∗ ) = G ∗ 2 < 1.

(30)

Therefore, (21a) (21b)

Combining Eq. (18a) and (18b), we have u(s+1) = Gu(s) + b,

(λ − bk )(λ − bk+1 ) − ck ak+1 = 0,

(22)

Hence, the convergence follows. Now we discuss the TAGE algorithm, when N is odd. For simplicity, let us denote dk = bk + ρ1 , ek = bk − ρ1 , pk = bk +ρ2 and qk = bk −ρ2 , then the TAGE method Eqs. (18a)

133

TAGE METHOD FOR MIXED BOUNDARY VALUE PROBLEMS

Since (G 1 + ρ1 I) and (G 2 + ρ2 I) are non-singular, hence simplifying Eqs. (31a) and (31b), we obtain the following TAGE algorithms: First sweep: For k = 0(2)N − 1, let 1 = dk dk+1 −ck ak+1 = 0

and (18b) in matrix form may be written as: ⎡

d0


⎢a ⎢ 1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ⎡

0

c0 d1 d2 a3

c2 d3 ..

..

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

. dN−1 aN

⎡

⎤(s)

v0

⎢v ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎥ ×⎢ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢v ⎥ ⎣ N−1 ⎦ vN ⎡

.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ cN−1 ⎥ ⎦ dN ⎤(s)

RH 0 − e0 u0

⎢ RH − e u − c u ⎥ 1 1 1 1 2 ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ =⎢ ⎥ . .. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ RH ⎥ N−1 − aN−1 uN−2 − eN−1 uN−1 ⎦ ⎣ RH N − eN uN 0

p0 p1 a2

c1 p2 ..

. ..

. pN−2 aN−1

cN−2 pN−1

0 ⎡

u0

⎤

⎤

(31a)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ u ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢u ⎥ ⎣ N−1 ⎦ uN

⎡

RH 0 − q0 v0 − c0 v1

⎢ RH − a v − q v 1 1 0 1 1 ⎢ ⎢ .. ⎢ ⎢ . ⎢ ⎢ .. ⎢ . ⎢ ⎢ =⎢ .. ⎢ . ⎢ ⎢ .. ⎢ ⎢ . ⎢ ⎢ RH N−1 − qN−1 vN−1 − cN−1 vN ⎣ RH N − aN vN−1 − qN vN

(s) vk+1

= (S2 dk − S1 ak+1 )/1 ,

Second sweep: For k = 0, we have u(s+1) = RH 0 − q0 v0(s) − c0 v1(s) /p0 , 0

(32) (33)

s = 0, 1, 2 ........ (34)

For k = 1(2)N − 2, let 2 = pk pk+1 − ck ak+1 = 0 (s) − qk vk(s) , S3 = RH k − ak vk−1 (s) (s) − ck+1 vk+2 , S4 = RH k+1 − qk+1 vk+1

then u(s+1) = (S3 pk+1 − S4 ck )/2 , k

s = 0, 1, 2 . . . . . . .,

(35)

u(s+1) k+1

s = 0, 1, 2 . . . . . .

(36)

= (S4 pk − S3 ak+1 )/2 ,

s = 0, 1, 2 . . . . . .

⎤(s) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (31b)

s = 0, 1, 2 . . .

vk(s) = (S1 dk+1 − S2 ck )/1 ,

Finally for k = N, we have (s) u(s+1) = RH N − aN vN−1 − qN vN(s) /pN , N

pN ⎤(s+1)

(s) S1 = RH k − A∗ ak u(s) k−1 − ek uk , (s) ∗ S2 = RH k+1 − ek+1 uk+1 − B ck+1 u(s) k+2 ,

0, if k = 0 where A∗ =

1, otherwise,

0, if k = N − 1 and B ∗ =

1, otherwise, then

(37)

In a similar manner, we can write the TAGE algorithm when N is even and the Newton-TAGE method for the non-linear difference equation (see Mohanty et al. [6]). 4. NUMERICAL ILLUSTRATIONS To illustrate our method, we consider two examples of mixed boundary value problems. We have compared the proposed TAGE and Newton-TAGE iterative methods with the corresponding SOR and Newton–SOR iterative methods (see [9–12]). In all cases, we have considered u(0) = 0 and the iterations were stopped when |u(s+1) − u(s) | ≤ 10−10 was achieved. In all cases, we have estimated the optimal values of ρ, ρ1 , and ρ2 (ρopt , ρ1 opt and ρ2 opt ). Throughout our computation, we choose σk = σ = a constant, k = 1(1)N, then the value of first mesh spacing on the left is given by h1 =

(b − a)(1 − σ ) , σ = 1 (1 − σ N )

(38)

134


and SOR methods are tabulated in Table 1 for σ = 0.8 and σ = 1.2. Example 2: (Nonlinear Problem)

TABLE 1 The RMS errors SOR Method


N σ = 0.8 11 21 31 41 51 61 σ = 1.2 11 21 31 41 51 61

ρopt

iter

TAGE Method ρ1opt

ρ2opt

iter

RMS errors

u(0) − u (0) = 0, 1.507 1.679 1.738 1.759 1.778 1.793

40 65 94 114 120 131

0.492 0.285 0.230 0.195 0.169 0.155

0.488 33 0.282 59 0.215 79 0.178 96 0.162 96 0.145 107

0.4715 (-03) 0.2036 (-03) 0.1583 (-03) 0.1364 (-03) 0.1219 (-03) 0.1113 (-03)

1.531 1.699 1.761 1.789 1.803 1.812

40 66 89 102 109 118

0.592 0.344 0.255 0.218 0.182 0.139

0.586 33 0.3216 (-02) 0.340 60 0.1623 (-02) 0.250 81 0.1262 (-02) 0.201 99 0.1084 (-02) 0.182 104 0.9688 (-03) 0.139 114 0.8842 (-03)

Thus by prescribing the total number of grid points (N + 1), we can determine √the remaining mesh √ sides by hk+1 = σ hk , k = 1(1)N-1, where 5 − 1 < 2σ < 5 + 1. Example 1: (Linear Problem) u = 4x 3 u + 12x 2 u, 0 ≤ x ≤ 1,

(39)

u(0) − 2u (0) = 1, u(1) + 2u (1) = 9e.

(40a) (40b) 4

The exact solution is given by u(x) = ex . The root mean square (RMS) errors and number of iterations both for TAGE TABLE 2 The RMS errors Newton-SOR Method N σ = 0.8 11 21 31 41 51 61 σ = 1.2 11 21 31 41 51 61

ν u = uu + f (x), 0 ≤ x ≤ 1,

Newton-TAGE Method

ρopt

iter

ρ1opt

ρ2opt

ter

RMS errors

1.420 1.613 1.715 1.748 1.768 1.783

8 13 19 22 25 28

0.0560 0.0294 0.0240 0.0195 0.0174 0.0155

0.0552 0.0283 0.0236 0.0191 0.0164 0.0150

7 10 14 17 20 22

0.1101 (-02) 0.5470 (-03) 0.4325 (-03) 0.3734 (-03) 0.3339 (-03) 0.3047 (-03)

1.510 1.684 1.752 1.785 1.804 1.812

10 16 19 21 24 25

0.0640 0.0340 0.0254 0.0200 0.0185 0.0173

0.0634 0.0334 0.0245 0.0194 0.0179 0.0166

9 13 16 19 20 21

0.3752 (-03) 0.2011 (-03) 0.1577 (-03) 0.1357 (-03) 0.1212 (-03) 0.1106 (-03)

(Burgers equation) (41) (42a)

−1

u(1) + u (1) = 2e + e .

(42b)

The exact solution is given by u(x) = x 2 cosh x. The right hand side function f (x) can be obtained by using the exact solution as a test procedure. The RMS errors and number of iterations both for Newton-TAGE and Newton-SOR methods are tabulated in Table 2 for ν = 0.1, σ = 0.8 and σ = 1.2. 5. CONCLUSIONS We have presented an efficient third-order accurate variable mesh finite difference discretization for the solution of two point nonlinear boundary value problems with linear mixed boundary conditions. The application of TAGE and Newton–TAGE iterative methods has been successfully implemented. The efficiency is achieved by including in the algorithm two optimal relaxation parameters, ρ1 and ρ2 , which are tested numerically to be accurate. The effect of the TAGE and Newton –TAGE iterative methods on non-uniform mesh is demonstrated by solving two test problems. It is hoped that the proposed technique will be useful for the development of new parallel algorithms for the solution of nonlinear partial differential equations. REFERENCES 1. H.B. Keller, Numerical Methods for Two-point Boundary Value Problems, Blaisdell, New York, 1968. 2. M.M. Chawla, A Fourth-order Tri-Diagonal Finite Difference Method for General Two-point Boundary Value Problems with Mixed Boundary Conditions, J. Inst. Math. Applics, vol. 21, pp. 83–93, 1978. 3. M.K. Jain, S.R.K. Iyengar, and G.S. Subramanyam, Variable Mesh Methods for the Numerical Solution of Two Point Singular Perturbation Problems, Comp. Meth. Appl. Mech. Engg., vol. 42, pp. 273–286, 1984. 4. D.J. Evans and R.K. Mohanty, On the Application of the SMAGE Parallel Algorithms on a Non-uniform Mesh for the Solution of Non-linear Two Point Boundary Value Problems with Singularity, Int. J. Computer. Math., vol. 82, pp. 341–353, 2005. 5. R.K. Mohanty, A Family of Variable Mesh Methods for the Estimates of (du/dr) and Solution of Non-linear Two Point Boundary Value Problems with Singularity, J. Comp. Appl. Math., vol. 182, pp. 173–187, 2005. 6. R.K. Mohanty and Noopur Khosla, Application of TAGE Iterative Algorithms to an Efficient Third-Order Arithmetic Average Variable Mesh Discretization for Two Point Nonlinear Boundary Value Problems, Appl. Math. Comput., vol. 172, pp. 148–162, 2006. 7. D.J. Evans, Group Explicit Method for Solving Large Linear Systems, Int. J. Computer. Math., vol. 17, pp. 81–108, 1985. 8. D.J. Evans, Iterative Methods for Solving Non-linear Two Point Boundary Value Problems, Int. J .Computer. Math., vol. 72, pp. 395–401, 1999. 9. D.M. Young, Iterative Solution of Large Linear Systems, Dover Publications, New York, 2003. 10. R.S. Varga, Matrix Iterative Analysis, Springer Verlag, Berlin, 2000. 11. L.A. Hageman and D.M. Young, Applied Iterative Methods, Dover Publications, New York, 2004. 12. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM Publications, Philadelphia, 2003.