A NUMERICAL SCHEME FOR THE PARABOLIC

c 2006 Institute for Scientific ° Computing and Information

INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 2, Number 3, Pages 326–335

A NUMERICAL SCHEME FOR THE PARABOLIC EQUATION SUBJECT TO MASS SPECIFICATION M. AKRAM AND M. A. PASHA Abstract. In this paper an O(h4 +l4 ) L0 -stable parallel algorithm is developed for the solution of the parabolic partial differential equation ut = uxx , 0 < x < 1, 0 < t ≤ T , subject to u(x, 0) = f (x), 0 < x < 1, ux (1, t) = g(t), 0 < t < T, R and the specification of mass 0b u(x, t)dx = m(t), 0 < b < 1. Key Words. Parabolic Equation; Mass Specification; Rational Approximation; Parallel Algorithm.

1. Introduction It is known that the nonlocal problems are very important in the transport of reactive and passive contaminates in aquifer, an area of active interdisciplinary research of mathematicians, engineers, and scientists [4, 5]. Mathematical formulation of this kind also arises naturally in various engineering models, such as nonlocal reactive transport in underground water flows in porous media [4, 6], heat conduction [21, 22], radioactive nuclear decay in fluid flows , semiconductor modeling and biotechnology [20]. This paper addresses the problem of obtaining numerical approximations to the u(x, t) which satisfies the parabolic partial differential equation(PDE): (1)

ut = uxx , 0 < x < 1, t > 0

subject to a Neumann time-dependent boundary condition on boundary ∂R of the open region R defined by lines x = 0 and x = X given by (2)

ux (1, t) = g(t), 0 ≤ t ≤ T,

and the specification of mass Z b (3) u(x, t)dx = m(t), 0 < b < 1, 0

with initial condition (4)

u(x, 0) = f (x), 0 < x < 1,

where f (x), g(t), b, and m(t) are known, while the function u(x, t) is to be determined. J. R. Cannon et al. [1] studied the existence and uniqueness properties of this problem. A number of sequential numerical schemes have been suggested in the literature for the solution of the problem [2,3]. A. B. Gumel [12] developed O(h2 + l2 ) L0 -stable parallel algorithm for this problem, and found to be more Received by the editors 10 February, 2005. 2000 Mathematics Subject Classification. 65M06, 65M20, 68W10. This research was supported by Punjab University College of Information Technology, Pakistan. 326

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accurate in comparison with two existing algorithms [2, 3]. For this problem, M. Akram and M. A. Pasha [19] developed O(h3 +l3 ) L0 -stable parallel algorithm, and found to be more accurate in comparison with [12]. This paper presents O(h4 + l4 ) L0 -stable parallel algorithm for this problem. The comparison of numerical results clearly demonstrates the computational superiority of this parallel algorithm with existing algorithms. In this paper, the method of lines semidiscretization approach is used to transform the model partial differential equations (PDEs) into a system of first-order linear ordinary differential equations (ODEs), the solution of which satisfies a certain recurrence relation involving matrix exponential terms. A suitable rational approximation will be used to approximate such exponentials leadings to a L0 -stable algorithm which may be parallelized through a partial-fraction splitting scheme. These L0 -stable schemes, unlike A0 -stable schemes are known to be suitable for use in integrating PDEs with time-dependent boundary conditions in which discontinuities between initial condition and boundary conditions exist [23]. The paper is organized in the following way: the numerical scheme is described in Section 2; parallel algorithm is also presented in Section 3; In Section 4, the numerical results produced by this method are compared with those demonstrated in [12, 19]; the conclusion is given in Section 5. 2. Discretization and recurrence relation The interval 0 ≤ x ≤ 1 is divided into N + 1 subintervals each of width h, so that (N + 1)h = 1 and the time variable t is discretized in the steps of length l. Thus at each time level t = tn = nl(n = 0, 1, 2, · · · ), the open region R = {0 < x < 1} × [t > 0] and its boundary ∂R consisting of lines x = 0 and x = 1 and the axis t = 0 have been superimposed by rectangular mesh with N points within R and open point along each side of ∂R. The solution u(x, t) of (1) is sought at each point (kh, nl) in R × [t > 0], where k = 1, 2, · · · , N and n = 0, 1, 2, · · · . The solution of an approximating numerical method will be denoted by U(x, t). The space derivative in (1) and (2) will be replaced by their fourth-order central-difference approximations given by ∂ 2 u(x, t) ∂x2 (5)

= +

1 {−u(x − 2h, t) + 16 u(x − h, t) − 30 u(x, t) 12h2 16 u(x + h, t) − u(x + 2h, t)} + O(h6 ) as h → 0

and ∂u(x, t) ∂x (6)

1 {u(x − 2h, t) − 8u(x − h, t) + 8u(x + h, t) 12h u(x + 2h, t)} + O(h4 ) as h → 0.

= −

The integral in (3) may be approximated using a quadrature rule such as Simpson’s rule to give Z

b

u(x, t)dx ≈ 0

(7)

+

n/2−1 n/2 X X b {u(0, t) + 2 u(2i, t) + 4 u(2i − 1, t) 3n i=1 i=1

u(n, t)} + O(h4 ).

Thus the boundary conditions u(1, t) and u(0, t) may then be determined using (2) with (6) and (3) with (7) respectively. Applying (1) to the mesh points of the

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interval [0, 1] at time level t = nl produces a system of ODE’s of the form dU(t) = AU(t) + v(t), t > 0 dt

(8) with the initial condition (9)

U(0) = f

in which A is a matrix of order n of the form  a1 a2 a3 a4 · · ·  16 −30 16 −1   −1 16 −30 16 −1   −1 16 −30 16 1  (10) A=  . . . 2 .. .. .. 12h    −1 16   −1 °

an −1 .. .

°

..

. −30 16 16 −30 −2 24



         −1   16  −30

where a1 = −6, a2 = −1 and   −4 for i = 3(2)n − 1, −2 for i = 4(2)n − 2, ai :=  −1 for i = n. In (8), vector v(t) aries from use of boundary conditions u(0, t) and u(1, t) in (5). Solving (8) subject to (9) gives Z t (11) U(t) = exp(lA)u(0) + exp((t + l − s)A)v(s)ds; 0

which satisfies the recurrence relation Z t+l (12) U(t + l) = exp(lA)U(t) + exp((t + l − s)A)v(s)ds; t = 0, l, 2l, . . . . t

in which l is a constant time step in the discretization of the time variable t ≥ 0 at the points tn = nl(n = 0, 1, 2, · · · , N ). To approximate the matrix exponential in (12), we consider rational approximation to exp(lA) of the form (13)

exp(lA) =

in which b1 = 1 − a1 , b2 = Thus (14) µ

I + b1 lA + b2 l2 A2 + b3 l3 A3 I − a1 l A + a2 l2 A2 − a3 l3 A3 + a4 l4 A4 1 2

− a1 + a2 ,b3 =

1 6

− a21 + a2 − a3 , a4 =

−1 24

+ a61 − a22 + a3 .

1 1 a1 exp(lA) = G−1 I + (1 − a1 )lA + ( − a1 + a2 )l2 A2 + ( − + a2 − a3 )l3 A3 2 6 2

¶

where G = I − a1 l A + a2 l2 A2 − a3 l3 A3 + a4 l4 A4 .

(15)

The denominator of exp(lA) has distinct real zeros choice choosing the values a1 = 2.70, a2 = 2.58, a3 = 1.02 and L-stability is studied in [15]. The integral term in (12) will be approximated by quadrature formula of the form Z t+l (16) exp((t + l − s)A)v(s)ds = W1 v(s1 ) + W2 v(s2 ) + W3 v(s3 ) + W4 v(s4 ) t

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where all si (i = 1, 2, 3, 4) are different. Matrices W1 , W2 , W3 and W4 can be obtained [15]. ª l © 3I − (19 − 78a1 + 216a2 − 324a3 )lA + (3 − 8a1 + 12a2 )l2 A2 G−1 W1 = 24 ª 3l © W2 = 2I + (16 − 56a1 + 144a2 − 216a3 )lA + (1 − 4a1 + 12a2 − 24a3 )l2 A2 G−1 16 ª 3l © W3 = I − (7 − 26a1 + 72a2 − 108a3 )lA − (1 − 4a1 + 12a2 − 24a3 )l2 A2 G−1 8 l © W4 = 6I + (44 − 168a1 + 432a2 − 648a3 )lA + (11 − 44a1 + 132a2 − 216a3 )l2 A2 48 ª + (2 − 8a1 + 24a2 − 48a3 )l3 A3 G−1 . Hence (12) may be written as l 2l (17) U(t + l) = exp(lA)U(t) + W1 v(t) + W2 v(t + ) + W3 v(t + ) + W4 v(t + l). 3 3 3. The parallel algorithm We focused on the construction of a rational approximation with real and distinct poles. The algorithm readily admits parallelization through partial fraction expansion [11, 13, 15]. Let r1 , r2 , r3 and r4 be distinct real zeros of the denominator of exp(lA). Then l l l l A)(I − A)(I − A)(I − A) r1 r2 r3 r4 and the approximation to the matrix exponential function may be written in partial fraction form as ½ l l l exp(lA) = p1 (I − A)−1 + p2 (I − A)−1 + p3 (I − A)−1 r1 r2 r3 ¾ l + p4 (I − A)−1 , r4 G = (I −

in which pj , the partial fraction coefficients of exp(lA) are defined by pj =

1 + (1 − a1 )rj + ( 21 − a1 + a2 )rj2 + ( 16 − Q4 rj i = 1 (1 − ri ) i 6= j

a1 2

+ a2 − a3 )rj3

,

j=1, 2, 3, 4. The implementation of the scheme using a parallel architecture with four processors is based on the partial fraction decomposition of exp(lA)U(t), W1 v(t), W2 v(t + 3l ), W3 v(t + 2l 3 ) and W4 v(t + l) in (17). ½ l l l l p5 (I − A)−1 + p6 (I − A)−1 + p7 (I − A)−1 W1 v(t) = 24 r1 r2 r3 ¾ l + p8 (I − A)−1 v(t), (18) r4 where p4+j =

3 + (−19 + 78a1 − 216a2 + 324a3 )rj + (3 − 8a1 + 12a2 )rj2 , Q4 rj i = 1 (1 − ri ) i 6= j

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M. AKRAM AND M. A. PASHA

j=1,2,3,4 l W2 v(t + ) 3 (19)

= +

3l 16

½

l l l A)−1 + p10 (I − A)−1 + p11 (I − A)−1 r1 r2 r3 ¾ l l p12 (I − A)−1 v(t + ), r4 3 p9 (I −

where p8+j =

2 + (16 − 56a1 + 144a2 − 216a3 )rj + (1 − 4a1 + 12a2 − 24a3 )rj2 , Q4 rj i = 1 (1 − ri ) i 6= j

j=1,2,3,4 2l W3 v(t + ) = 3 (20)

+

3l 8

½

l l l A)−1 + p14 (I − A)−1 + p15 (I − A)−1 r1 r2 r3 ¾ l 2l p16 (I − A)−1 v(t + ) r4 3 p13 (I −

where p12+j =

1 + (7 + 26a1 − 72a2 + 108a3 )rj − (1 − 4a1 + 12a2 − 24a3 )rj2 , Q4 rj i = 1 (1 − ri ) i 6= j

j=1,2,3,4 and W4 v(t + l) (21)

= +

l 48

½

l l l A)−1 + p18 (I − A)−1 + p19 (I − A)−1 r1 r2 r3 ¾ l p20 (I − A)−1 v(t + l) r4 p17 (I −

where p16+j

1

=

Q4

+

ª (11 − 44a1 + 132a2 − 216a3 )rj2 + (2 − 8a1 + 24a2 − 48a3 )rj3 ,

i = 1 (1 − i 6= j

rj ri )

{6 + (44 − 168a1 + 432a2 − 648a3 )rj

j=1,2,3,4. Hence equation (17) gives the following form (22)

U(t + l) = y1 (t) + y2 (t) + y3 (t) + y4 (t)

PARABOLIC EQUATION SUBJECT TO MASS SPECIFICATION

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in which y1 , y2 , y3 and y4 are the solutions of the systems ½ 2l l l A1 y1 = p1 U(t) + 2p5 v(t) + 9p9 v(t + ) + 18p13 v(t + ) 48 3 3 (23) + p17 v(t + l)} , ½ l l 2l A2 y2 = p2 U(t) + 2p6 v(t) + 9p10 v(t + ) + 18p14 v(t + ) 48 3 3 (24) + p18 v(t + l)} , ½ l 2l l A3 y3 = p3 U(t) + 2p7 v(t) + 9p11 v(t + ) + 18p15 v(t + ) 48 3 3 (25) + p19 v(t + l)} , ½ l l 2l A4 y4 = p4 U(t) + 2p8 v(t) + 9p12 v(t + ) + 18p16 v(t + ) 48 3 3 + p20 v(t + l)} . (26) Equations (23-26) have great importance in the parallel environment since they can be used to solve the corresponding linear algebraic systems on processors operating simultaneously. U (t + l) in (22), the solution vector at time t = (n + 1)l, may now be obtained via the parallel algorithm using four different processors in the following form: Processor 1 (1) Input l, r1 , U(0), A (2) Compute p1 , p5 , p9 , p13 , p17 , I − (3) Decompose I −

l r1 A

l r1 A

= L1 U1

(4) Evaluate v(t), v(t + 3l ), v(t +

2l 3 ),

v(t + l)

(5) use z1 (t) = 2p5 v(t) + 9p9 v(t + 3l ) + 18p13 v(t + (6) Solve L1 U1 y1 (t) = p1 U(t) + Processor 2 (1) Input l, r2 , U(0), A

l r2 A

+ p17 v(t + l)

l 48 z1 (t)

(2) Compute p2 , p6 , p10 , p14 , p18 , I − (3) Decompose I −

2l 3)

l r2 A

= L2 U2

(4) Evaluate v(t), v(t + 3l ), v(t +

2l 3 ),

v(t + l)

(5) use z2 (t) = 2p6 v(t) + 9p10 v(t + 3l ) + 18p14 v(t + (6) Solve L2 U2 y2 (t) = p2 U(t) + Processor 3 (1) Input l, r3 , U(0), A

l 48 z2 (t)

(2) Compute p3 , p7 , p11 , p15 , p19 , I −

l r3 A

2l 3)

+ p18 v(t + l)

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M. AKRAM AND M. A. PASHA

(3) Decompose I −

l r3 A

= L3 U3

(4) Evaluate v(t), v(t + 3l ), v(t +

2l 3 ),

v(t + l)

(5) use z3 (t) = 2p7 v(t) + 9p11 v(t + 3l ) + 18p15 v(t + (6) Solve L3 U3 y3 (t) = p3 U(t) +

2l 3)

+ p19 v(t + l)

2l 3)

+ p20 v(t + l)

l 48 z3 (t)

Processor 4 (1) Input l, r4 , U(0), A (2) Compute p4 , p8 , p12 , p16 , p20 , I − (3) Decompose I −

l r4 A

l r4 A

= L4 U4

(4) Evaluate v(t), v(t + 3l ), v(t +

2l 3 ),

v(t + l)

(5) use z4 (t) = 2p8 v(t) + 9p12 v(t + 3l ) + 18p16 v(t + (6) Solve L4 U4 y4 (t) = p4 U(t) +

l 48 z4 (t)

GOTO step 4 for next time step. Hence U(t + l) = y1 (t) + y2 (t) + y3 (t)+ y4 (t). In implementing the parallel algorithm, Processor 1 generates once only decomposition of I − rl1 A, while Processor 2 generates once only decomposition of I − rl2 A, Processor 3 generates once only decomposition of I − rl3 A and Processor 4 generates once only decomposition of I − rl4 A simultaneously.

4. Results and Discussions In order to test the behavior of L0 -stable parallel algorithm, four examples from the literature are considered. The parallel algorithm is tested on a sequential computer (Intel. 933 MHz, BD815 Glly, 128MB(Kingstung), HDD 20 GB (SeaCate), OS Win2000 Professional, Developer Stuido ) for the solutions of the parabolic partial differential equations. A comparison with [12, 19] for the model problem clearly demonstrates that the new technique is computationally superior. Example 3.1. Consider the parabolic partial differential equation with f (x) = g(t) =

0.5x2 1.0

m(t) =

1 0.75t + (0.75)3 6

The analytical solution is u(x, t) = 0.5x2 +t. The absolute relative error |(u−U )/u| computed results at various time lengths with h = l = 0.0025 are shown in Table 1.

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Table 1. Relative errors at various time lengths with h = l = 0.0025. Time length

O(h2 + l2 )[12] O(h3 + l3 )[19] O(h4 + l4 ) Analytical Solution

t=0.1

8.07E − 4

4.32E − 5

2.34E − 6

0.13125

t=0.025

8.81E − 5

5.75E − 6

4.12E − 7

5.625E − 2

t=0.010

1.61E − 6

1.02E − 7

1.13E − 9

4.125E − 2

Example 3.2. Consider the parabolic partial differential equation with f (x) =

sin(x)

g(t) =

−πe−π t 2 1 1 (p + 1)e−π t π (2)

m(t) = u(x, t) =

2

2

e−π t sin(πx)

This example has an analytical solution u(x, t) = e−t cos(x). The computed results at various time lengths with h = l = 0.0025 are shown in Table 2. Table 2. Relative errors at various time lengths with h = l = 0.0025. O(h2 + l2 )[12]

O(h3 + l3 ) [19] O(h4 + l4 )

Analytical Solution

t=0.1

6.29E − 5

2.30E − 6

2.11E − 7

0.8767

t=0.025

2.82E − 6

2.13E − 7

1.11E − 8

0.9450

t=0.010

3.73 − 8

2.31E − 9

1.21E − 10

0.9593

Time length

Example 4.3. Consider the parabolic partial differential equation with f (x) =

sin(x)

g(t) =

−πe−π t 2 1 1 + 1)e−π t (p π (2)

m(t) =

2

2

This example has an analytical solution u(x, t) = e−π t sin(πx). The computed results at various time lengths with h = l = 0.0025 are shown in Table 3. Example 4.4. Following the effect of time-steps on the O(h4 +l4 ) parallel algorithm is investigated by solving Example 4.3 with three different time-steps, namely l = 0.01, l = 0.05 and l = 0.025. The relative errors at t = 0.1 are given in Table 4.

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M. AKRAM AND M. A. PASHA

Table 3. Relative errors at various time lengths with h = l = 0.0025. O(h2 + l2 )[12]

O(h3 + l3 ) [19] O(h4 + l4 )

Analytical Solution

t=0.1

3.55E − 5

6.35E − 7

1.13E − 8

0.2635

t=0.025

2.81E − 7

8.65E − 9

2.42E − 10

0.5525

t=0.010

2.95E − 6

1.43E − 8

7.68E − 10

0.6407

Time length

Table 4. Relative errors at t = 0.1 with h = 0.0025 using various time-steps. O(h2 + l2 )[12]

O(h3 + l3 )[19]

O(h4 + l4 )

Analytical Solution

l=0.01

3.99E − 4

3.23E − 5

2.36E − 6

0.2635

l=0.05

1.28E − 2

7.65E − 4

3.02E − 5

0.2635

l=0.025

3.355E − 5

2.43E − 6

1.11E − 7

0.2635

Time step

Clearly, discontinuities between initial conditions and boundary conditions exist in all four problems. Tables 1 to 4 confirm that our L0 -stable scheme is more accurate then existing ones. 5. Conclusion An O(h4 + l4 ) L0 -stable scheme has been tested on model problems presented in the literature related to parabolic partial differential equations with Neumann time-dependent boundary condition. The developed algorithm works efficiently for such problems. The comparison of the results demonstrated the computational superiority of the new scheme. References [1] J. R. Cannon and J. van der Hoek , The existance and a continuous dependence result for the heat equation subject to the specification of engery, Suppl. Bollettino Unione Mat. Ital.1(1981), 253-282. [2] J. R. Cannon , Y. Lin and S. Wang, An implicit finite difference scheme for the diffusion equation subject to mass specification , Int.J. Engng. 28 (1990), no. 7, 573-578. [3] J. R. Cannon , S. Perez Esteva , and van der Hoek J., A Galekin procedure for the diffusion equation subject to the specication of mass, SIAM J. Numer. Anal.24, no. 3(1987), 499-515. [4] J. H. Cushman and T. R. Ginn, Nonlocal dispersion in porous media with continuously evolving scales of heterogeneity, Transp. Porous Media 13 (1993), 123-138. [5] J. H. Cushman , H. Xu and F. Deng, Nonlocal reactive transport with physical and chemical heterogeneity: localization error,Water Resources Res. 31 (1995), 2219-2237.

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[6] G. Dagan, The significance of heterogeneity of evolving scales to transport in porous formations, Water Resources Res. 13 (1994), 3327-3336. [7] M. Dehghan , On the numerical solution of the diffusion equation with a nonlocal boundary condition , Mathematical Problems in Engineering ,2 (2003) 81-92. [8] G. Ekolin , Finite difference methods for a nonlocal boundary value problem for the heat equation, BIT 31 (1991), no. 2, 245-261. [9] A. Friedman , Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401-407. [10] G. Fairweather and R. D. Saylor , The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. Sci. Statist. Comput. 12 (1991), no. 1, 127-144. [11] A. R. Gourlay and J. Morris , The extrapolation of first order methods for parabolic partial differential equations. II, SIAM J. Numer. Anal. 17(1980), no. 5, 641-655. [12] A. B. Gumel, On the numerical solution of the diffusion equation subject to the specification of mass, J. Austral, Math. Soc, ser. B 40 (1999), 475-483. [13] J. D.Lambert, Numerical methods for ordinary differential systems. The Initial Value Problem, John Wiley and and Sons, Chichester, 1991. [14] M. Sapagoves , Hypothesis on the solability of parabolic equations with nonlocal conditions , Nonlinear Analysis: Modelling and control, 7(2002), 93-104. [15] M. S. A. Taj and E. H. Twizell , A family of fourth-order parallel splitting methods for parabolic partial differential equations, Methods partial differential equations 13, no.4 (1997) 357-373 . [16] S. Wang and Y.Lin , A numerical method for the diffusion equation with nonlocal boundary specifications , Internat. J. Engrg. Sci. 28 (1990), 543-546. [17] Y.Lin , An inverse problem for a class of quasilinear parabolic equation , SIAM J. Math. Anal. 22(1), (1991), 146-156. [18] V. A. Vodakhova, A boudnary-value problem with Nakhushev non-local condition for certain pseudo-parabolic water-transfer equation, Differentsialnie Uravnenia, 18(1982), 280-285. [19] M. Akram and M. A. Pasha, A numerical method for the heat equation with a nonlocal boundary condition, IJISS, 1(2)(2005), 162-171. [20] W. Allegretto, Y. Lin and A. Zhou, , A box scheme for coupled systems resulting from microsensor thermistor problems, Dynam. Contin. Discrete Impuls. Systems 5 (1999), 209223. [21] J. R. Cannon, The solution of the heat equation subject to the specification of engery , Quart. Appl. Math., 21(1963), 155-160. [22] N. I. Ionkin,Solution of boundary-value problem in heat-conduction theory with nonclassical boundary condition, Differentsialine Uravenia, 13(1977), 1177-1182. [23] M. S. A. Taj and E.H. Twizell., A family of third-order parallel splitting methods for parabolic partial differential equations, Intern. J. Computer Math., Vol. 67, pp. 411-433. [24] A. M. Nakhushev, On certain approximate method for boundary-value problems for differntial equations and its applicjations in ground waters dynmaics, Differentsialnie Uravenia, 18 (1982), 280-285. M. Akram is an Assistant Professor at Punjab University College of Information Technology, University of the Punjab, Pakistan. His research areas of interest are Numerical Solutions of Parabolic/Hyperbolic PDEs and (Fuzzy) Algebras. He has published many research papers in these areas. E-mail: [email protected] M. A. Pasha received his Ph.D. degree in Computer Science from the University of Southampton, U. K. in 1996. His research areas of interest are Knowledge Management, Intelligent Information Systems, Data Warehouse, Parallel Algorithms and Fuzzy Systems. He has published many research articles and has 25 years of postgraduate teaching experience. Presently, he is working as Principal, Punjab University College of Information Technology, University of the Punjab, Pakistan. E-mail: [email protected]