New Group Iterative Schemes In The Numerical Solution Of The TwoDimensional Time Fractional Advection-Diffusion Equation Alla Tareq Balasim and Norhashidah Hj. Mohd. Ali Accepted Manuscript Version This is the unedited version of the article as it appeared upon acceptance by the journal. A final edited version of the article in the journal format will be made available soon.
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Publisher: Cogent OA Journal: Cogent Mathematics DOI: https:// doi.org/10.1080/23311835.2017.1412241
New Group Iterative Schemes In The Numerical Solution Of The Two-Dimensional Time Fractional Advection-Diffusion EquationI Alla Tareq Balasima,b,∗, Norhashidah Hj. Mohd. Alia a School
of Mathematical Sciences, Universiti Sains Malaysia, 11800, Penang, Malaysia of Mathematics, College of Basic Education, University of Al-mustansiria, Iraq
b Department
Abstract Numerical schemes based on small fixed-size grouping strategies have been successfully researched over the last few decades in solving various types of partial differential equations where they have been proven to possess the ability to increase the convergence rates of the iteration processes involved. The formulation of these strategies on fractional differential equations, however, is still at its infancy. Appropriate discretization formula will need to be derived and applied to the time and spatial fractional derivatives in order to reduce the computational complexity of the schemes. In this paper, the design of new group iterative schemes applied to the solution of the 2D time fractional advectiondiffusion equation are presented and discussed in detail. The Caputo fractional derivative is used in the discretization of the fractional group schemes in combination with the Crank-Nicolson difference approximations on the standard grid. Numerical experiments are conducted to determine the effectiveness of the proposed group methods with regard to execution times, number of iterations and computational complexity. The stability and convergence properties are also presented using a matrix method with mathematical induction.The numerical results will be proven to agree with the theoretical claims. Keywords: Group methods, Two-dimensional problem, Time fractional ∗ Alla
Tareq Balasim Email address:
[email protected] (Alla Tareq Balasim)
Preprint submitted to Cogent Mathematics
October 9, 2017
advection-diffusion, Caputo fractional derivative, Crank-Nicolson difference schemes, Stability & Convergence analysis.
1. Introduction Fractional calculus, which is the calculus of integrals and derivatives in random order, dates back as far as the more popular integer calculus, and has been gaining significant interest over the past few years with its history and 5
development being explored in detail by ([1], [2], [3] and [4]). Fractional differential equations (FDEs) can be used to model many problems in a wide field of applications. They are defined as equations that utilize fractional derivatives and considered as powerful tools that can describe the memory and hereditary characteristics of various materials. Several researchers have explored the use of
10
FDEs in the fields of chemistry ([5, 6]), physics ([7, 8, 9, 10]) and other scientific and engineering spheres ([11, 12, 13, 14]). Since there are at most no exact solutions to the majority of fractional differential equations, it is necessary to resort to approximation and numerical methods [15, 16, 17, 18]. Over the past decade, there has been an influx of numerical methods development for solving various
15
types of FDEs ([19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]). Chen et al. [27] used implicit and explicit difference techniques to solve time fractional reaction-diffusion equations, while Liu et al. [28] proposed for these techniques to be employed in solving the space-time fractional advectiondispersion equation by replacing the first-order time derivative by the Caputo
20
fractional derivative, and the first-order and second-order space derivatives by the Riemman-Liouville fractional derivatives. The use of radial basis function (RBF) approximation method was discussed in Uddin et al. [29] to solve the time fractional advection-dispersion equation in a bounded domain. The application of finite element method was also seen in Zheng et al. [30], in solving
25
the space fractional advection-diffusion equation under non-homogeneous initial boundary conditions. In Chen et al. [31], a numerical method with first-order temporal accuracy and second-order spatial accuracy was established for solving
2
the variable order Galilei advection-diffusion equation with a nonlinear source term, while Zhuang et al. [32] used the explicit and implicit Euler approximation 30
to solve a variable order fractional advection-diffusion equation with a nonlinear source term. A new numerical solution for the two-dimensional fractional advection-dispersion equation with variable coefficients in a finite field was also introduced by Chen and Liu [33]. In 2011, Shen et al. [34] proposed the explicit and implicit finite difference approximations for the space-time Riesz-Caputo
35
fractional advection-diffusion equation where the implicit scheme was proven to be unconditionally stable, while the explicit scheme exhibits a conditionally stable property. The development of an implicit meshless method was also seen in Zhuang et al. [35] in solving the time-dependent fractional advection-diffusion equation where a discretized system of equations was obtained using the moving
40
least squares (MLS) meshless shape functions. In general, finding numerical solutions to FDEs using iterative finite difference schemes is not a straightforward process and can be a challenging task due to several reasons. Firstly, all the earlier solutions have to be saved if the current solution is to be computed, making the calculations even more com-
45
plex and costly in terms of CPU usage time in cases where traditional point implicit difference approaches are used. Secondly, although the point implicit techniques are stable, a significant amount of CPU usage time is required when many unknowns are involved. In recent decades, grouping strategies have been proven to possess characteristics that are able to reduce the spectral radius of
50
the generated matrix resulting from the finite difference discretization of the differential equation, and therefore increase the convergence rates of the iterative algorithms. They have been shown to reduce the computational timings compared to their point wise counterparts in solving several types of partial differential equation ([36, 37, 38, 39, 40, 41, 42]). These group methods are also
55
easy to implement and suitable to be implemented on parallel computers due to their explicit nature. However, till date these strategies have not been tested on solving FDEs, particularly for the two-dimensional cases. Therefore, in this study, the formulation of new group iterative methods is presented in solving 3
the following two-dimensional time fractional advection diffusion equation ∂αu ∂2u ∂2u ∂u ∂u = ax 2 + ay 2 − bx − by + f (x, y, t) , α ∂t ∂x ∂y ∂x ∂y
(1)
where 0 < α < 1, ax , ay , bx , by are positive constants and f (x, y, t) is nonhomogeneous term subjected to the following initial and Dirichlet boundary conditions u(x, y, 0) = g(x, y) u(0, y, t) = g1 (y, t) u(1, y, t) = g2 (y, t) u(x, 0, t) = g3 (x, t) u(x, 1, t) = g4 (x, t)
60
This equation plays an important role in describing transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns [35]. This paper is outlined as follows. Section 2 presents the proposed group iterative methods obtained from the Crank-Nicolson difference approximation followed by the stability and convergence analysis of the differ-
65
ence schemes in Sections 3 and 4 respectively. Section 5 presents the discussion on the computational effort involved in solving equation (1) using the proposed methods with regard to the arithmetic operations for each iteration. Finally, the results of the numerical experiments are presented and discussed in Section 6
70
2. Standard approximation schemes for fractional advection-diffusion equations The Caputo fractional derivative, Dα , of the order-α is expressed as follows [43].: 1 D = Γ(m − α) α
Zx
f m (t)
α−m+1 dt,
0
(x − t)
α > 0, m − 1 < α < m, m ∈ N, x > 0, (2)
4
where Γ(.) is the Euler Gamma function. Further details on the definitions and 75
properties of fractional derivatives are available in [4]. We need to apply appropriate finite difference approximations to the time and space derivaties of (1), let h > 0 be the space step and k > 0 be the time step. The domain is assumed to be uniform in both x and y directions. Define xi = ih, yj = jh, {i, j = 0, 1, ..., n}, and a mesh size of h =
80
1 n,
where n is an
arbitrary positive integer and tk = kτ, k = 0, 1, ..., l. various approximations formulas could be obtained for (1) at the point of (xi , yj , tk ). By taking the average of the central difference approximations to the left side of (1) at the points (i, j, k + 1) and (i, j, k), the Caputo time fractional approximation (2) can be transformed to the following form [44] ∂ α u(xi ,yj ,tk+1 ) ∂tα
= w1 uk +
+O(τ
2−α
Pk−1 s=1
[wk−s+1 − wk−s ] usi,j − wk u0i, j + σ
k (uk+1 i,j −ui,j )
21−α
), (3)
85
where σ =
1 τ α Γ(2−α)
, ws = σ((s +
1 1−α 2)
− (s −
1 1−α ). 2)
Using (3) in combination with the second order Crank-Nicolson difference scheme for the right side of (1) will result in the following fractional standard point (FSP) formula Pk−1 (uk+1 −uk ) w1 uk + s=1 [wk−s+1 − wk−s ] usi,j − wk u0i, j + σ i,j21−α i,j k+1 k+1 k k ui−1,j −2uk+1 uk ax i,j +ui+1,j i−1,j −2ui,j +ui+1,j = 2 + h2 h2 k+1 k+1 k+1 k+1 k k k ui,j−1 −2ui,j +ui,j+1 ui+1,j −uk+1 uk ui,j−1 −2uk ay bx i,j +ui,j+1 i−1,j i+1,j −ui−1,j +2 + − 2 + h2 h2 2h 2h k+1 k+1 k k 1 ui,j+1 −ui,j−1 u −u k+ b 2 2 − 2y + i,j+12h i,j−1 + fi, j 2 + O(τ 2−α + (4x) + (4y) ). 2h (4)
5
Equation (4) can be simplified to become as follows sx (1 + sx + sy ) uk+1 i,j = ( 2 + sy 2 sx +( 2
+(
s
c
k+1 y y + ( s2x − c4x )uk+1 i+1,j + ( 2 + 4 )ui,j−1 c 1−α ∗ − 4y )uk+1 w1 − sx − sy uki,j + ( s2x + c4x )uki−1,j i,j+1 + 1 − 2 s
k+1 cx 4 )ui−1,j
c
s
c
− c4x )uki+1,j + ( 2y + 4y )uki,j−1 + ( 2y − 4y )uki,j+1 + 21−α wk∗ u0i,j s Pk−1 ∗ k+1/2 ∗ +21−α s=1 wk−s − wk−s+1 ui,j + m0 fi,j (5) 90
1−α α
t Γ(2 − α),
where m0 = 2 sx =
ax m 0 h2 , sy
=
ay m0 h2 ,
cx =
ws∗
= (s + 1/2)
bx m 0 h ,
cy =
1−α
− (s − 1/2)
1−α
,
by m 0 h .
The figure below shows the computational molecule for equation (5).
i,j+1,k+1
sy /2-cy /4
i+1,j,k+1
i,j,k+1
i-1,j,k+1
sx/2-cx/4
1+sx+sy
sx/2+cx/4
i+1,j,k i, j-1,k+1
Time level k+1
sy /2-cy /4
sy /2+cy /4 i-1,j,k
i+1,j,k
i,j,k
sx/2+cx/4
sx/2-cx/4
1-w1 -sx-sy
Time level k
i,j-1,k sy /2+cy /4
i,j,k-1 Time level (w1 -w2 ) k-1
i,j,k-2 Time level
(w2 -w3 )
k-2 . . i,j,1 . Time level
(wk-1 -wk )
1 i,j,0
Time level
wk
0
Figure 1: Computational molecule of the standard point approximation.
6
2.1. Fractional Explicit Group Method In formulating the fractional explicit group (FEG) method, (5) is applied to 95
any group of four points in the solution domain to generate a 4 × 4 system of equation as follows
D
−a2 0 −b2
−a1
0
D
−b1
−b2
D
0
−a1
−b1
ui,j
−a2 D 0
=
ui+1,j ui+1,j+1 ui,j+1
rhsi,j rhsi+1,j rhsi+1,j+1 rhsi,j+1
,
(6)
where D = 1 + sx + sy , a1 = ( s2x − c4x ), b1 = (
sy 2
−
cy 4 ),
a2 = ( s2x + c4x ), b2 = (
sy 2
+
cy 4 ),
R = (1 − 21−α w1∗ − sx − sy ), k+1 k k k k rhsi,j =a2 (uk+1 i−1,j + ui−1,j ) + b2 (ui,j−1 + ui,j−1 ) + a1 ui+1,j + b1 ui,j+1
+2
1−α
wk∗ u0i,j
+
Ruki,j
+
k−1 X
k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi.j + m0 fi,j
,
s=1 k+1 k k k k rhsi+1,j =a1 (uk+1 i+2,j + ui+2,j ) + b2 (ui+1,j−1 + ui+1,j−1 ) + b1 ui+1,j+1 + a2 ui,j
+ 21−α wk∗ u0i+1,j + Ruki+1,j +
k−1 X
k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi+1,j + m0 fi+1,j ,
s=1
rhsi+1,j+1 =a1 (uk+1 i+2,j+1
+
uki+2,j+1 )
k k k + b1 (uk+1 i+1,j+2 + ui+1,j+2 ) + b2 ui+1,j + a2 ui,j+1
+ 21−α wk∗ u0i+1,j+1 + Ruki+1,j+1 +
k−1 X
k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi+1,j+1 + m0 fi+1,j+1 ,
s=1
rhsi,j+1 =a2 (uk+1 i−1,j+1
+
uki−1,j+1 )
+
b1 (uk+1 i,j+2
+ 21−α wk∗ u0i,j+1 + Ruki,j+1 +
k−1 X s=1
7
+ uki,j+2 ) + a1 uki+1,j+1 + b2 uki,j k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi,j+1 + m0 fi,j+1 .
Mathematical software can be used to easily invert (6) to obtain the FEG formula
r 1 ui+1,j 1 r5 = ui+1,j+1 r r7 ui,j+1 r8 ui,j
r2
r3
r1
r4
r8
r1
r9
r2
r4
r6 r5 r1
rhsi,j
rhsi+1,j , rhsi+1,j+1 rhsi,j+1
(7)
where 1 4 (cx + c4y + 8c2y 4 + 5s2x + 8sy + 3s2y + 8sx (1 + sy ) + 16[9s4x + 48s3x (1 + sy ) 256 2 + 4 + 8sy + 3s2y + 2s2x 44 + 88sy + 39s2y + 16sx 4 + 12sy + 11s2y + 3s3y ] +c2x −2c2y + 8 4 + 3s2x + 8sy + 5s2y + 8sx (1 + sy ) ), 1 = (1 + sx + sy ) c2x + c2y + 4 4 + 3s2x + 8sy + 3s2y + 8sx (1 + sy ) , 16 1 =− (cx − 2sx ) c2x − c2y + 4 4 + 3s2x + 8sy + 5s2y + 8sx (1 + sy ) , 64 1 = (cx − 2sx ) (cy − 2sy ) (1 + sx + sy ) , 8 1 =− (cy − 2sy ) −c2x + c2y + 4 4 + 5s2x + 8sy + 3s2y + 8sx (1 + sy ) , 64 1 = (cx + 2sx ) c2x − c2y + 4 4 + 3s2x + 8sy + 5s2y + 8sx (1 + sy ) , 64 1 = − (cx + 2sx ) (cy − 2sy ) (1 + sx + sy ) , 8 1 = (cx + 2sx ) (1 + sx + sy ) (cy + 2sy ) , 8 1 = (cy + 2sy ) −c2x + c2y + 4 4 + 5s2x + 8sy + 3s2y + 8sx (1 + sy ) , 64 1 = − (cx − 2sx ) (1 + sx + sy ) (cy + 2sy ) . 8
r=
r1 r2 r3 r4 r5 r6 r7 r8 r9
Figure 2 shows the construction of blocks of four points in the solution domain 100
for the case n=10. Note that if n is even, there will be ungrouped points near the upper and right sides of the boundary. The FEG method proceeds with the iterative evaluation of solutions in these blocks of four points using equation (7) throughout the whole solution domain until convergence is achieved. For the case of even n, the solutions at the ungrouped points near the boundaries are
8
105
computed using (5).
ϭϬ ϵ ϴ
j
ϳ 6 5 4 3 2 1 i 0 1 2 3 4
5
6 7
8
9 10
Figure 2: Grouping of the points for the FEG method (n=10).
2.2. The fractional modified explicit group method In this new method, we consider the nodal points with grid size spacing 2h = 2 n.
The standard fractional formula is generated through the application specific
finite difference approximations with 2h-spaced points. Using the Caputo time 110
fractional approximation (3) at the left hand side of (1) and the second order Crank-Nicolson difference scheme with 2h-spaced points at the right hand side of (1), the following approximation formula is obtained: Pk−1 (uk+1 −uk ) w1 uk + s=1 [wk−s+1 − wk−s ] usi,j − wk u0i, j + σ i,j21−α i,j k+1 k+1 k k ui−2,j −2uk+1 uk i,j +ui+2,j i−2,j −2ui,j +ui+2,j = a2x + 4h2 4h2 k+1 k+1 k+1 k+1 k k k ui,j−2 −2ui,j +ui,j+2 ui+2,j −uk+1 ui,j−2 −2uk uk a bx i,j +ui,j+2 i−2,j i+2,j −ui−2,j + 2y − + + 2 4h 4h 4h2 4h2 k+1 k+1 k k ui,j+2 −ui,j−2 u −u k+1/2 b 2 2 − 2y + i,j+24h i,j−2 + fi, j + O(τ 2−α + (M x) + (M y) ). 4h (8)
9
On simplification, we obtain the following 1+
sx + sy sx cx sx cx sy cy uk+1 + )uk+1 + ( − )uk+1 + ( + )uk+1 i,j = ( 4 8 8 i−2,j 8 8 i+2,j 8 8 i,j−2 sy cy sx + sy sx cx +( − )uk+1 + 1 − 21−α w1∗ − uki,j + ( + )uki−2,j 8 8 i,j+2 4 2 8 sx cx k sy cy k sy cy k +( − )ui+2,j + ( + )ui,j−2 + ( − )ui,j+2 + 21−α wk∗ u0i,j 8 8 8 8 8 8 k−1 X s k+1/2 ∗ ∗ +21−α wk−s − wk−s+1 ui,j + m0 fi,j . s=1
(9) Applying (9) to any group of 4 points (i,j), (i+2,j), (i+2,j+2) and (i,j+2) in the 115
solution domain will result in the following 4 × 4 system
D1
−a22 0 −b22
−a11
0
−b11
D1
−b11
0
−b22
D1
−a22
0
−a11
D1
ui,j
ui+2,j ui+2,j+2 ui,j+2
=
rhsi,j rhsi+2,j rhsi+2,j+2 rhsi,j+2
, (10)
where, D1 = s ( 8y
+
cy 8 ),
s +s (1+ x 4 y ),
a11 =
Q = (1 − 21−α w1∗ −
( s8x − c8x ), sx +sy 4 ),
b11 =
s ( 8y
c − 8y ),
a22 =
( s8x + c8x ),
b22 =
k+1 k k k k rhsi,j =a22 (uk+1 i−2,j + ui−2,j ) + b22 (ui,j−2 + ui,j−2 ) + a11 ui+2,j + b11 ui,j+2
+ 21−α wk∗ u0i,j + Quki,j +
k−1 X
k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi.j + m0 fi,j
,
s=1 k+1 k k k k rhsi+2,j =a11 (uk+1 i+4,j + ui+4,j ) + b22 (ui+2,j−2 + ui+2,j−2 ) + b11 ui+2,j+2 + a22 ui,j
+ 21−α wk∗ u0i+2,j + Quki+2,j +
k−1 X s=1
10
k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi+2,j + m0 fi+2,j ,
k+1 k k k k rhsi+2,j+2 =a11 (uk+1 i+4,j+2 + ui+4,j+2 ) + b11 (ui+2,j+4 + ui+2,j+4 ) + b22 ui+2,j + a22 ui,j+2
+ 21−α wk∗ u0i+2,j+2 + quki+2,j+2 +
k−1 X
k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi+2,j+2 + m0 fi+2,j+2 ,
s=1
rhsi,j+2 =a22 (uk+1 i−2,j+2
+
uki−2,j+2 )
+
b11 (uk+1 i,j+4
+ 21−α wk∗ u0i,j+2 + Quki,j+2 +
k−1 X
+ uki,j+4 ) + a11 uki+2,j+2 + b22 uki,j k+1/2
∗ ∗ 21−α [wk−s − wk−s+1 ]usi,j+2 + m0 fi,j+2 .
s=1
The four-point FMEG equation below is obtained by inverting (10), as follows
r 11 k+1 ui+2, j r = 1 55 k+1 const ui+2, j+2 r77 k+1 ui, j+2 r88
cons =
uk+1 i.j
r22
r33
r11
r44
r88
r11
r99
r22
r44
r66 r55 r11
rhsi,j
rhsi+2,j , rhsi+2,j+2 rhsi,j+2
(11)
1 (4096 + c4x + c4y + 4096sx + 1408s2x + 192s3x + 9s4x + 4096sy 4096
+3072sx sy + 704s2x sy + 48s3x sy + 1408s2y + 704sx s2y + 78s2x s2y + 192s3y +48sx s3y + 9s4y + 2c2y (64 + 5s2x + 32sy + 3s2y + 8sx (4 + sy )) +2c2x 64 − c2y + 3s2x + 32sy + 5s2y + 8sx (4 + sy ) ), 1 (4 + sx + sy ) 64 + c2x + c2y + 32sx + 3s2x + 32sy + 8sx sy + 3s2y , 256 1 =− (cx − sx ) 64 + c2x − c2y + 32sx + 3s2x + 32sy + 8sx sy + 5s2y , 512 1 =− (cx − sx ) 64 + c2x − c2y + 32sx + 3s2x + 32sy + 8sx sy + 5s2y , 512 1 =− (cy − sy ) 64 − c2x + c2y + 32sx + 5s2x + 32sy + 8sx sy + 3s2y , 512 1 = (cx + sx ) 64 + c2x − c2y + 32sx + 3s2x + 32sy + 8sx sy + 5s2y , 512 1 =− (cx + sx ) (cy − sy ) (4 + sx + sy ) , 128 1 = (cx + sx ) (cy + sy ) (4 + sx + sy ) , 128
r11 = r22 r33 r44 r55 r66 r77
11
1 (cy + sy ) 64 − c2x + c2y + 32sx + 5s2x + 32sy + 8sx sy + 3s2y , 512 1 =− (cx − sx ) (cy + sy ) (4 + sx + sy ) . 128
r88 = r99
To use this method, the grid points in the solution domain are divided into 3 types of points, denoted by the symbols , 4 and • in alternate ordering as shown in Figure 3. Note that the evaluation of (11) require points of type • 120
only. Thus, we can construct the FMEG method by generating the iterations on this type of points only, followed by the evaluation of solutions directly once on points of type and 4 . The FMEG algorithm can then be summarized as follows: 1-Divide the grid points into three types , 4 and • in alternate order as
125
depicted in Fig. 3. 2- Set the initial guess for the iterations. 3- Evaluate the solutions at points • using equation (11) iteratively at the time level k + 1. 4- Step 5 is performed if the iteration converges. Otherwise, Step 3 is repeated
130
until a convergence is attained. 5- The steps below are performed directly once for points and 4 in Figure 3 at the time level k + 1: a) For type points, the rotated h − spaced five-point approximation formula derived by rotating the x − y axis clockwise at 45o was used [40]. This
135
approximation formula was applied to the right side of Eq. (1), in combination with (3) being applied to the left side of (1), to obtain the following rotated C-N formula:
1+
+(
sy 4
sx +sy 2
c −c
c +c
s
c −cx
k+1 sx sx x y x y y y uk+1 )uk+1 i,j = ( 4 + i−1,j+1 + ( 4 − 8 )ui+1,j+1 + ( 4 + 8 8 cx +cy sx +sy cx −cy k+1 sx 1−α ∗ k k + 8 )ui−1,j−1 + 1 − 2 w1 − 2 ui,j + ( 4 + 8 )ui−1,j+1 c +c
s
c −c
+( s4x − x 8 y )uki+1,j+1 + ( 4y + y 8 x )uki+1,j−1 + ( s Pk−1 ∗ k+1/2 ∗ +21−α s=1 wk−s − wk−s+1 ui,j + m0 fi,j .
sy 4
+
cx +cy k 8 )ui−1,j−1
+ 21−α wk∗ u0i,j
b) For type 4 points, the typical h − spaced formula (5) is used. Note that
12
)uk+1 i+1,j−1
formula (5) involve points of type 4 only M L 0
1
2
3
4
5
6
7
8
9
10
Figure 3: Grouping of the points for the FMEG method (n=10).
140
3. Stability analysis A scheme is considered to be stable if the errors cease to increase with the passing of time, and gradually become inconsequential as the computation progresses. Even though the spacing is different, Equation (5) gives rise to both the FEG and FMEG methods. Therefore, the stability of both methods can be
145
analyzed in similar ways. Here, we will show the stability of the FMEG scheme using eigenvalues of the generated matrices with mathematical induction. Form (10) we obtain AU 1 = BU AU k+1 = BU k − C1 U k +
k=0 Xk−1 s=1
13
(12) Ck−s U s + Ck U o + b
k>0
where
R1
R2
R 3 A= 0
R1
0 R2 .. . R3
0
0
R1 R3
G1
G3
G 2 R1 =
G1
S1 S 0 3 ,B = 0 R2 R1 0 0
G3 .. . G2
G1 G2
G4
R3 = S2 =
G4 ..
. G4
H5 H5 ..
. H5
S2 S1
0
S1 S3
G5
, R2 = G3 G1 H1 H 2 , S1 = G4 H4 , S3 = H5
14
0
S2 .. . S3
W1 W 0 1 . , b = .. W1 S2 S1 W1
0
G5 ..
. G5 G5
H3 H1
H3 .. . H2
H1 H2
H3 H1
H4 ..
. H4 H4
Mk Ck =
, Ck−s
Mk ..
. Mk Mk
M k−s =
, Mk−s
Mk−s ..
. Mk−s
s = 1, ..., k − 1
150
M1
C1 =
M1 ..
. M1
0
0 G2 = 0 0
−b22
0
0
0
−b22
0
0
0
0
0 −a11 G5 = 0 0
0
0
0
0
0
0
0
0
L1 D 1 L 1 −a22 .. , W1 = . , G1 = 0 L1 −b22 M1 L1
0
0 0 , G3 = 0 0 0 −b11 0 Q a 0 , H1 = 22 0 −a11 0 b22
15
−a11
0
D1
−b11
−b22
D1
0
−a11
0 0 −a22 0 0 0 0 0 G4 = 0 −b11 0 0 0 0 0 0 0 0 a11 0 b11 0 0 0 0 Q b11 0 , H2 = 0 0 b22 Q a22 0 a11 Q 0 0 0
−b11
0 −a22 D1
0
0
0
0 0 −a22 0 b22 0 0
b22
0 , 0 0
0 0 0
0
0 H3 = 0 b11
0
0
0
0
b11
0
0
0
Mk−s =
21−α tα
M1 =
155
2
tα
a22
0
0
0
0
0
0
a22
0 0 a 0 , H5 = 11 0 0 0 0
0
0
0
0
0
0
0
0
0
0 a11 0
∗ ∗ (wk−s − wk−s+1 )
0
0
0
0
∗ ∗ (wk−s − wk−s+1 )
0
0
0
0
∗ ∗ (wk−s − wk−s+1 )
1−α
0 0 0 0 , H4 = 0 0 0 0
w∗ 1 0 0 0
0 0
0
0
∗
w1
0
0
w1∗
0
0
0
0 , Mk = 0 w1∗
2
1−α
tα
w∗ k 0 0 0
0
0
∗
wk
0
0
wk∗
0
0
0 ∗
∗ (wk−s − wk−s+1 )
0
0
0 , 0 wk∗
fi,j f i+2,j 1−α , L1 = 2 Γ[2 − α] fi+2,j+2 fi,j+2 i, j = 2, 6, ..., n − 2.
The following lemma is important to prove the stability.
Lemma 1: [44] In (12), the coefficients ws∗ , s = 1, 2, ..., k satisfy the following
160
∗ ∗ 1 − wk−s > wk−s+1 , s = 1, 2, ..., k. Pk−1 ∗ ∗ 2 − S=1 [wk−s − wk−s+1 ] + wk∗ = w1∗ , s = 1, 2, ..., k.
For simplicity we assume sx = sy = S = 1 tα
+
S 2
and Q =
1 tα
−2
1−α
Theorem 1: : If 165
w1∗
−
Γ(2−α)21−α , cx h2
−
S 2
−
Γ(2−α)21−α , D1 h
=
S 2
√ 1 tα
= cy = C =
21−α ∗ w1 tα
+
−C 2 +S 2 4
> 0 then the FMEG scheme
(10) is stable.
Proof : To prove the stability of (10) we suppose that uki,j , i, j = 1, 2, ..., n, k = k 1, 2, ..., l are the approximate solutions to the exact solution Ui,j of (1), the error
16
,
k εki,j = Ui,j − uki,j be the error at time level k. From (12), the error satisfies
AE 1 = BE
k=0
AE k+1 = BE k − C1 E k + 170
Xk−1 s=1
(13) Ck−s U s + Ck E o
k>0
where E1k+1 εk+1 2 E k+1 εk+1 1 4 . . E k+1 = .. , E1k+1 = .. , εk+1 = i k+1 k+1 E ε 1 m−4 E1k+1 εk+1 m−2 i = 2, 6, ..., n − 2, j = 2, 6, ..., n − 2.
εk+1 i,j
εk+1 i+2,j , εk+1 i+2,j+2 k+1 εi,j+2
From the above equations, the following are obtained: A = G4 + G2 + G1 + G3 + G5 B = H4 + H2 + H1 + H3 + H5 C1 = M1
(14)
Ck−s = Mk−s Ck = Mk It is worthy to note that, from (14), the eigenvalues of the matrices A, B, C1 , Ck−s and Ck are ak , bk , c1 , ck−s and ck respectively, where
p p 4 1 4 2 + S2 , 2 + S2 + 2S − −C + 2S + −C tα 4 tα p p 1 S 1 S 1 4 1 4 2 2 2 2 bk = − , α − , − 2S − −C + S , − 2S + −C + S tα 2 t 2 4 tα 4 tα ak =
1 S 1 S 1 + , α + , tα 2 t 2 4
21−α ∗ w1 tα 21−α ∗ ∗ ck−s = α (wk−s − wk−s+1 ) t 21−α ck = α wk∗ t c1 =
17
For k = 0 E 1 = A−1 B E 0 1 tα −
1
E ≤ ρ(A−1 B) E 0 = 1 + tα
√
S 2 S 2
E 0 √ 2 2 −C +S + 4 −C 2 +S 2 4
+
1 0
E ≤ E
∴ Supposing that:
kE s k ≤ E 0 , s = 1, ..., k.
Need to prove this inequality holds for s = k + 1. E k+1 = A−1 (B − C1 )E k +
k−1 X
A−1 Ck−s E s + A−1 Ck E 0
s=1
X
k−1
≤ ρ(A−1 (B − C1 )) E 0 + ρ(A−1 Ck−s ) E 0 + ρ(A−1 Ck ) E 0 s=1
Using Lemma 1, we get √ 1 −C 2 +S 2 S 21−α ∗
4 tα − 2 − tα w1 +
k+1
E 0 +
E
≤ √ −C 2 +S 2 1 S 1 + 2 + + tα 4 tα
21−α ∗ w1 tα
√
S 2
+
−C 2 +S 2 4
E 0
∴ E k+1 ≤ E 0
Therefore, under the conditions 175
√ 1 tα
−
S 2
−
21−α ∗ w1 tα
+
−C 2 +S 2 4
> 0 the FMEG 2
iterative scheme (10) is stable.
4. Convergence analysis Theorem 2: The finite difference FMEG scheme (10) is convergent and the follow √
k+1 −C 2 +S 2 2 2 −1 2−α 1 S 21−α ∗
≤ C (τ ing estimate hold e +(M x) +(M y) ) if − − w + > 1 k tα 2 tα 4 0.
Proof : Let us denote the truncation error at (xi , yj , tk+1 ) by Rk+1 . From 8, we
have, Rk+1/2 ≤ c(τ 2−α + (M x)2 + (M y)2 )
18
Define η k+1 = U (xi , yj , tk+1 ) − uk+1 i, j = 1, 2, .., n, k = 1, 2, ..., l and ek+1 = i,j T
(ek+1 , ek+1 , ..., ek+1 ) , using e0 = 0 , where 1 1 1 η2k+1 k+1 ηi,j η k+1 4 k+1 ηi+2,j .. k+1 k+1 , {i, j} = 2, 6, ..., n − 2. e1 = . , εi = k+1 ηi+2,j+2 k+1 η k+1 m−4 ηi,j+2 k+1 ηm−2 Substitution into (12) produces: 1
Ae1 = R 2 Aek+1 = Bek − C1 ek +
Xk−1 s=1
1
Ck−s es−1 + Rk+ 2 .
Using mathematical induction to prove the above theorem, set C0−1 = 1. For k = 0 1
Ae1 = R 2
1
e ≤ ρ(A−1 ) kRk ≤ C0−1 (τ 2−α + (M x)2 + (M y)2 ).
Assume that
k −1
e ≤ C
k+ 21
. k−1 R Need to prove it hold for s = k + 1 k−1 P 1 Aek+1 = (B − C1 )ek + Ck−s es−1 + Rk+ 2 s=1
k−1
1
k+1
k X
−1 ρ(A−1 Ck−s ) es−1 + ρ(A−1 ) Rk+ 2
e
≤ ρ(A (B − C1 )) e + s=1
√ −C 2 +S 2 1 S 21−α ∗
4 tα − 2 − tα w1 +
k+1 +
e
≤ √ −C 2 +S 2 1 S 1 + 2 + + tα 4 tα
21−α ∗ w1 tα
√
S 2
+
−C 2 +S 2 4
−1
k+ 1
Ck 2
R (15)
19
≤ Ck−1 Rk =(
21−α ∗ wk ) tα
−1
(τ 2−α + (M x)2 + (M y)2 ) kα k−α τ α
2−α + (M x)2 + (M y)2 ) 1−α 1−α (τ 21−α ((k + 12 ) − (k − 21 ) ) 1 = 1−α (τ 2−α + (M x)2 + (M y)2 ) 2 (1 − α)
=
Since
lim
k−α 1−α −(k− 1 ) 2
1−α
1 k→∞ (k+ 2 )
=
2
1 . 1−α
5. Computational complexity This section presents an analysis of the computational complexity with regard 180
to the three techniques described for solving (1), which is the number of arithmetic operations per iteration. For simplicity we assume sx = sy , cx = cy . Suppose m2 internal points exist within the solution, with m = n − 1, where n has an even mesh size, then the ungrouped points will be found close to the right/upper boundaries. Internal mesh points have two main types, namely, the iteration points that participate
185
in the iteration process and the direct points that are directly calculated once using the rotated and standard difference formulas following the attainment of the iteration convergence. Table 1 lists the number of internal mesh points for the three earlier methods, whereas Table 2 provides a summary of the number of arithmetic operations that are needed for each iteration and the direct solution following the convergence,
190
not only for the explicit group methods, but also for the fractional standard point (FSP) method.
Table 1: Number of various types of mesh points for the point and explicit group methods
Point types
FSP m2 -
Iterative group points Iterative ungrouped points
2
Number of points FEG FMEG (m − 1)2 2m-1 2
Total iterative points
m
m
Direct h-spaced rotated points Direct h-spaced standard points Total direct points
-
-
20
(m−1)2 4
(m−1)2 4 (m+1)2 4 m2 −1 4 3m2 +2m−1 4
Table 2: Computational complexity for the point and explicit group methods Per Iteration Additional Multiplication (14 + 13(k − 1))m2 (10 + (k − 1))m2
Methods SFP FEG
(15 + 13(k − 1))(m − 1)2 + (14 + 13(k − 1))(2m − 1)
(14 + (k − 1))(m − 1)2 + (10 + (k − 1))(2m − 1)
FMEG
((15+13(k−1))(m−1)2 ) 4
((14+(k−1))(m−1)2 ) 4
After Convergence Additional Multiplication -
-
(14 + 13(k − 1))×
(10 + (k − 1))×
(3m2 +2m−1) 4
(3m2 +2m−1) 4
6. Numerical experiments and discussion of results Several numerical examples are presented in this section to prove the effectiveness of the fractional explicit group methods in solving the 2-D TFADE (1) with a 195
Dirichlet boundary condition. A computer with Windows 7 Professional and Mathematica software having a Core i7 GHZ and 4 GB of RAM was used for conducting the experiments. Example 1: The time fractional initial boundary value problem below was considered [35]
200
∂αu ∂tα
=
∂2u ∂x2
+
∂2u ∂y 2
−
∂u ∂x
−
∂u ∂y
+ 0.5Γ (3 + α) t2 ex+y
where Ω = {(x, y) |0 ≤ x ≤ 1, 0 ≤ y ≤ 1} is the solution domain with the exact solution being t2+α ex+y . Example 2: The following time fractional advection-diffusion equation was also considered [45] 205
∂αu ∂tα
=
∂2u ∂x2
+
∂2u ∂y 2
−
∂u ∂x
−
∂u ∂y
+
t1−α (sin x+sin y) Γ(2−α)
+ t(cos x + sin x + cos y + sin y)
with the initial and boundary conditions are given as u (x, y, 0) = 0
210
u (0, y, t) = t sin y ,
u(1, y, t) = t (sin1 + sin y),
u (x, 0, t) = t sin x ,
u (x, 1, t) = t (sin x+sin1),
0 < t < 1,
0 < x, y < 1.
Different mesh sizes of 6, 10, 14 and 18 and various time steps, which satisfy the stability conditions, with a fixed relaxation factor (Gauss-Seidel relaxation scheme) of 1.0 were used to run the experiments. During all the experiments, the norm for the convergence criteria was ι∞ , while error tolerance was set at ε = 10−5 . Numerical 215
results were obtained using of the methods described in Section 2 for different values of α. The number of iterations, the execution time and the error analysis are presented
21
in Tables 3 & 4 and Figures 4 − 7 for the fractional point method and the fractional group methods. Figure 6 shows the execution times for various mesh sizes for both Examples 1 & 2. Its clear that when fractional group methods are used, the results 220
are just as accurate as the FSP method. From the results obtained, it is observed that the execution timings were reduced by as much as 20% and 80% of the FSP method when the FEG and FMEG methods were used respectively. In contrast to the other tested methods, the fractional group methods, specifically the FMEG method, took a least times to compute the solutions. As shown in Tables 3 & 4 and Figure 6
225
the time taken for FMEG to be executed was only approximately 15.6-32.5% of the FEG method, and was 12.3-23.9% of the FSP method. To gauge the computational complexity, obtaining an approximation of the amount of calculations involved for each method is necessary. The estimation for the amount of computational work involved was determined by arithmetic operations that were performed for a single
230
iteration (as outlined in Section 5). With the help of Tables 1 & 2, and based on the assumption that an approximately equal amount of time was needed for adding, subtracting, multiplying, dividing and assigning, a summary of the total number of operations involved in the iterative methods is presented in Table 5. This shows that the FMEG requires the least number of operations, thus confirming the theoretical
235
complexity analysis.
Table 3: Comparison of the number of iterations, Execution times, average and maximum error for different time step and mesh size at α = 0.75 and α = 0.95
h
Example 1 4t
Method FSP FEG FMEG
Time 9.18 7.25 1.13
α = 0.75 Ite. Ave 11 6.485E-4 8 6.458E-4 2 2.063E-3
Max 1.030E-3 1.029E-3 3.375E-3
Time 6.16 5.49 1.15
α = 0.95 Ite. Ave 8 6.125E-4 6 6.120E-4 2 1.986E-3
Max 9.761E-4 9.763E-4 3.252E-3
1/6
1/100
1/10
1/350
FSP FEG FMEG
350.37 367.80 54.55
11 8 4
2.480E-4 2.094E-4 6.899E-4
4.409E-4 3.807E-4 1.27E-3
195.31 172.19 54.30
6 5 4
2.581E-4 2.031E-4 6.709E-4
4.664E-4 3.681E-4 1.240E-3
1/14
1/850
FSP FEG FMEG
3924.31 2941.31 700.02
10 7 4
2.172E-4 1.346E-4 3.402E-4
4.353E-4 4.353E-4 6.539E-4
2032.86 2024.52 604.66
5 5 3
2.506E-4 9.522E-5 3.317E-4
5.002E-4 1.801E-4 6.372E-4
1/18
1/1620
FSP FEG FMEG
22323.3 17268.0 4330.99
9 6 4
2.923E-4 1.613E-4 2.208E-4
6.114E-4 3.212E-4 4.015E-4
12224.3 10499.4 3786.52
5 4 3
1.599E-4 7.361E-5 1.966E-4
3.242E-4 1.471E-4 3.887E-4
22
Table 4: Comparison of the number of iterations, Execution times, average and maximum error for different time step and mesh size at α = 0.75 and α = 0.95
h
Example 2 4t
Method FSP FEG FMEG
Time 7.81 6.53 1.17
α = 0.75 Ite. Ave 9 4.320E-4 7 4.359E-4 2 1.364E-3
Max 6.690E-3 6.759E-4 2.276E-3
Time 5.2 4.83 1.32
α = 0.95 Ite. Ave 6 4.326E-4 5 4.372E-4 2 1.369E-3
Max 6.702E-4 6.781E-4 2.283E-3
1/6
1/100
1/10
1/350
FSP FEG FMEG
294.69 230.39 55.13
9 6 4
1.173E-4 1.269E-4 5.190E-4
1.974E-4 2.192E-4 9.475E-4
169.2 155.22 48.40
5 4 3
1.126E-4 1.334E-4 5.205E-4
1.835E-4 2.238E-4 9.504E-4
1/14
1/850
FSP FEG FMEG
3267.46 2621.1 690.99
8 6 4
1.577E-5 5.451E-5 2.643E-4
2.441E-5 9.851E-5 4.999E-4
1772.2 1718.99 596.96
4 4 3
1.321E-5 6.546E-5 2.652E-4
3.995E-5 1.202E-4 5.017E-4
1/18
1/1620
FSP FEG FMEG
17647.9 13574.3 4351.54
7 5 4
5.607E-5 4.239E-6 1.584E-4
1.264E-4 1.0105E-5 3.057E-4
10034.0 9708.83 3764.74
4 3 3
1.780E-5 1.494E-5 1.591E-4
4.500E-5 3.291E-5 3.072E-4
Table 5: Total computing operations involved for the point and grouping methods
Method FSP FEG FMEG
Total operations Example 1 α = 0.75 α = 0.95 196 625 143 000 143 704 107 778 23 002 23 002
Total operations Example2 α = 0.75 α = 0.95 160 875 107 250 125 741 89 815 23 002 23 002
1/350
FSP FEG FMEG
2 196 315 1 600 140 341 462
1 197 990 1 000 090 341462
1 796 985 1 200 100 341 462
998 325 800 068 301 934
1/14
1/850
FSP FEG FMEG
10 080 850 7 062 140 1 766 910
5 040 425 5 044 390 1 551 970
8 064 680 6 053 260 1 766 910
4 032 340 4 035 510 1 551 970
1/18
1/1620
FSP FEG FMEG
29 534 355 19 698 000 5 828 950
16 711 425 13 374 800 5 196 200
26 229 420 18 743 200 6 672 190
14 988 240 11 245 900 5 836 220
h
4t
1/6
1/100
1/10
Conclusion This paper presented the development and formulation of new fractional explicit group iterative methods for solving the 2D-TFADE. The C-N difference schemes with h spacing and 2h spacing gave rise to the FEG and FMEG methods, respectively. The 240
stability and convergence of the proposed methods were analyzed using the matrix form with mathematical induction. Through the numerical experiments, the FMEG method stood out amongst all the other tested methods when it comes to its execution time and number of iterations, as it requires the least number of operation counts. It is noted that in terms of accuracy, the FMEG method is just as good as the FSP
23
245
method and the FEG method. This work confirms the suitability and feasibility of the grouping strategies in solving the two-dimensional time fractional advection-diffusion equation. The implementation of similar group methods on solving other fractional differential equations will be reported soon.
t 1
t 1
2.6
2.6
2.4
2.4
2.2
2.2
FSP
FEG FM EG
2.0
2.0
1.8
FM EG
1.8
1.6
1.6
x 0.2
0.3
0.4
0.5
0.6
0.7
x
0.8
0.2
0.3
0.4
(a)
0.5
0.6
0.7
0.8
(b)
Figure 4: A comparison of the numerical solution of the example 1 at α = 0.75, y=1/6, t=1 between,(A) FSP and FMEG (B) FEG and FMEG
t 1
t 1
0.9
0.9
0.8
0.8
0.7
0.7
FSP
FEG
FM EG
0.6
0.6
0.5
FM EG
0.5
0.4
0.4
x
x 0.2
0.3
0.4
0.5
0.6
0.7
0.2
0.8
(a)
0.3
0.4
0.5
0.6
0.7
0.8
(b)
Figure 5: A comparison of the numerical solution of the example 2 at α = 0.75, y=1/6, t=1 between,(A) FSP and FMEG (B) FEG and FMEG
24
Time
Time 20000
15 000
FSP
FSP
FEG
FEG
15000 FM EG
FMEG
10 000
10000
5000 5000
mesh size
mesh size 6
8
10
12
14
16
18
6
8
10
12
(a)
14
16
18
(b)
Figure 6: Experimental results for the mesh size against the Time at α = 0.75, (A) Example 1, (B) Example 2.
ite
ite 9
10
8 7
8
6
FSP
6
FSP
FEG
5
FEG
4
FM EG
FM EG 4 3
mesh size 8
10
12
14
16
mesh size 8
18
10
12
(a)
14
16
18
(b)
Figure 7: Experimental results for the mesh size against the Ite. at α = 0.75, (A) Example 1, (B) Example 2.
ACKNOWLEDGMENTS 250
Financial support provided by the Research University Individual (RUI) Grant (1001/PMATHS/8011016) for the completion of this article is gratefully acknowledged
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PUBLIC INTEREST STATEMENT Many phenomena in science and engineering can be modelled as two dimensional fractional advection-diffusion equations satisfying appropriate initial and boundary conditions. With the advent of computing technology, effective numerical methods have been extensively formulated in solving these equations due to their simplicity and accuracy. In particular, finite difference formulas, are oftenly used to numerically discretize the differential equations to produce sparse systems of linear equations which may be suitably solved by iterative solvers. However, iterative solvers have the disadvantage of being too slow to converge which may increase the computation timings. To overcome this problem, suitable small fixed-size grouping strategies are applied to the mesh points of the solution domain following the discretization of the differential equation which results in schemes with faster convergence and lower computational complexity with comparable accuracies. This is very useful to engineers and scientists who are involved in time consuming simulation modeling processes.
