Gen. Math. Notes, Vol. 18, No. 2, October, 2013, pp. 104-119
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Compact Finite Difference Methods for the
Solution of One Dimensional Anomalous SubDiffusion Equation
Faoziya S. Al-Shibani1, Ahmed I. Md. Ismail2 and Farah A. Abdullah3
1, 2, 3
School of Mathematical Sciences
Universiti Sains Malaysia, 11800 Penang, Malaysia
1
E-mail: fsmm11_mah020@student.usm.my
2
E-mail: izani@cs.usm.my
3
E-mail: farahaini@usm.my
(Received: 9-8-13 / Accepted: 14-9-13)
Abstract
In this paper, two high order compact finite difference schemes are formulated for
solving the one dimensional anomalous subdiffusion equation. The GrünwaldLetnikov formula is used to discretize the temporal fractional derivative. The
truncation error and stability of the two methods are discussed. The feasibility of
the compact schemes is investigated by application to a model problem.
Keywords:Compact finite difference method, Time fractional diffusion,
Grünwald-Letnikov method.
1
Introduction
Diffusionis a phenomenathat has been rigorously and extensively studied. This
phenomenon is modelled bythe diffusion equation which is a partial differential
equation that describes the spread of a substance whose particles move, due to the
random motion, from a region of higher concentration to a region of lower
Compact Finite Difference Methods for the...
105
concentration. The dependent variable in the equation is concentration of the
substance and the independent variables are spatial and temporal variables. The
diffusion equation is derived using Fick’s law which assumes a homogenous
environment. In the case of non-homogenous environments (for example, in a
cell) then the fractional diffusion equation is used to model the diffusion
process.Anomalous diffusion describes the spread of a particle plume at a rate
incompatible with the classical Brownian motion mode.Furthermore, the plume
may be asymmetric.When a cloud of particles spreads slower than classical
diffusion, it leads to anomalous subdiffusion [2].
The study of fractional partial differential equations has increased in recent years.
A comprehensive background on this topic can be found in books by Das[14] and
Podlubny[6]. Compact finite difference schemes are sometimes preferred because
their accuracy and high computational efficiency.Several papers have recently
been published on compact finite difference methods for solving theanomalous
diffusion equation. Gao and Sun [5] have presented a high order compact finite
difference scheme for the fractional subdiffusion equations.They first transformed
the problem usingthe Caputo definition and other analytical theories then thel1
discretization was applied to approximate the temporal fractional derivative. The
stability and convergence of the method were analyzed by the energy method. A
compact finite difference method was also discussed by Cui [8] for the
anomaloussubdiffusion equation. The Grünwald-Letnikov formulawas used to
approximate the time fractional derivative. The second order spatial derivative in
the problem was approximated with fourth order accuracy. The stability and the
convergence of the proposed method were investigated.Richardson extrapolation
was appliedto increase the accuracy in time as the generating function used in the
Grünwald-Letnikovformula givesfirst order accuracy in time.It was pointed out
that the use of the "Short Memory" principlemakes the solution inefficient for the
given example.Du et al.[11]developed a compact finite difference scheme for the
time fractional diffusion equation when 1 < < 2 so that the fractional wave
equation was also considered. The Caputo definition was applied to discretize the
time fractional derivative. The stability and convergence of the scheme were
analyzed in ∞ norm by using the energy method. It was found that the scheme is
unconditionally stable and has O(τ α + h ) convergence.
The purpose of this article is to develop two high order compact finite difference
methods for solving the one-dimensional fractional anomalous sub-diffusion
equation:
∂ α u ( x, t ) ∂ 2 u ( x, t )
=
+ f ( x, t ) , ( x, t ) ∈ (0,1) × (0, T ]
∂t α
∂x 2
(1)
With the initial condition
( , 0) = 0, 0 <
< 1
(2)
106
Faoziya S. Al-Shibani et al.
And the boundary conditions
(0, ) =
( ),
(1, ) =
( ), 0 ≤ ≤
(3)
and
are known functions, and the function u is the unknown
Where ,
function to be determined.We consider the case when 0 < α < 1 as we wish to
study the anomalous subdiffusion equation.We shall take h =1/Min the x direction
with xi = ih, i = 0,1 …M, tn= n τ and n = 0, 1…N.High order discretizations
require an additional number of grid points which induces more computational
effort. The two compact finite difference methods which will be described in this
article overcome these difficulties by involving derivatives of the function values
at the nodes of the corresponding independent variables. We eventually eliminate
the derivatives to get formulas with three points. This then leads to block
tridiagonal systems.
2
Fractional Derivatives and Integrals
In this section we give some relevant background on fractional derivatives and
related integral. Let fbe a function with respect to one independent variable t, the
α
fractional derivative 0 Dt of f(t)can be defined by Riemann-Liouville formula as
(see [12] ).
0
Dt α f (t ) =
t f (ι ) dι 
d  1
0<

dt  Γ(1 − α ) ∫0 (t − ι )α 
<1
Where Γ ( ) is the Gamma function and 0 ≤ ≤
(4)
.
The above derivative is related to the Riemann-Liouville fractional integral, which
is defined as
α
0 It
f (t ) =
1 t f (ι )dι
0<
Γ(α ) ∫0 (t − ι )1−α
<1
(5)
Where 0 D tα 0 I tα f ( t ) = f ( t )
(6)
Fractional derivatives can also be represented by the Grünwald-Letnikovformula
definedas ( see [10] ):
α
0
Dt f (t ) =
1
τ
α
t 
τ 
∑= ω α
( )
k
f (t − kτ ) + O(τ p ) ,
t≥0
k 0
α + 1 (α )
Where τt is integer and k = 0, 1, 2… τt and ω0(α ) = 1 , ω k(α ) = (1 −
)ω k −1 .
k
(7)
Compact Finite Difference Methods for the...
107
The coefficients ωk(α ) are the coefficients of the power series of the generating
function ω ( z , α ) = (1 − z )α and are also the coefficients of the two-point backward
difference approximation of the first order derivative.
 k − α − 1  k ∞ (α ) k
 z = ∑ ωk z
k

k =0 
k =0
∞
where
(1 − z )α = ∑ 
(8)
It is possible for the function ω ( z , α ) to generate coefficients of the high order
approximation, for instance if the generationg function is
ω ( z,α ) = (
25
12
− 4 z + 3z 2 −
4
3
z3 +
1
4
z 4 )α
(9)
then we get the Grünwald-Letnikovformula of a fourth order accuracy (i.e., P= 4)(
see [6] ).
The coefficients ωk(α ) can be expressed in term of the Fourier transform ( see [6] ).
If z = e − iϕ in Eq.(2.5) then ωk(α ) =
1
2π i
2π
fα (ϕ )e − ikϕ dϕ and fα (ϕ ) = (1 − e − iϕ )α .
∫
0
3
Compact Finite Difference
Grünwald-Letnikov Formula
Method
with
the
Let us define the central, forward and backward difference operators
=
"
!
$
"
#!
, %
We can represent
)
(
=
=
&
&
(
,
' =
+ ℎ %& ' +
&
− ℎ %& ' +
&
"
,%
%& + ' +
%& + ' −
($)+ & +
!
$
"
&#
about ( ,
($)+ & +
!
!
&
' =
%) & '
%& '
&
(
$
"
#!
%& - ' + ⋯
($)- & !
%& - ' + ⋯
($)- & !
(10)
) by Taylor seriesas (see [15] )
Analogously, the first and second derivatives of
&
"
(
,
0
(11)
are
5
= % & ' + ℎ % & + ' + ! % & - ' + ! % & 1 ' + ⋯3
4
&
&+
($)+ & ($)- & 1
= % & ' − ℎ % & + ' + ! % & - ' − ! % & 1 ' + ⋯3
2
&
&+
($)+ & -
($)- & 1
(12)
108
%
)
&+
&
%
'
+
'
& +
&+
Faoziya S. Al-Shibani et al.
=%
(
&+
&
=%
' +ℎ%
+
'
& +
&+
&&
−ℎ%
' +
-
'
& -
&-
+
%
($)+ & 1
!
&
%
'
& 1
($)+ & 1
!
' +
1
−
%
'
& 6
($)- & 6
!
4
% 6' + ⋯ 3
&
2
($)- & 6
!
+⋯ 5
3
(13)
We approximate the first and second spatial derivativesusing Eq.(11), (12) and
(13)toget
% ' =
&
&
78 "
9 (
:+ < < #
=
; <8 <8
+ >(ℎ? )
(14)
And
%
'
& +
&+
=
< <# @ "
%
'
<8 <8
+
: < <#
9 (
=
!+ <8 <8
+ >(ℎ? )
(15)
To obtain the approximate solution of Eq.(1), we replace the second order
derivative in space by the finite difference approximation in Eq.(15) and discretize
the time fractional derivative by using Grünwald-Letnikovdefinition (7).
When p = 1. We obtain
n

∂ +  ∂ −u 


1 n
 α ∑ ωk(α )uin − k = ∂x 2 ∂x i + f i n ,
 h ∂+ ∂− 
τ k = 0
 1 + 12 ∂x ∂x 




i = 1, 2,..., M − 1, ....n = 1, 2, ..., N ,
 0
ui = 0, ....i = 1, 2,..., M − 1,
 n
n
u0 = f1 (tn ),....u M = f 2 (tn ), ....n = 1, 2,..., N .
(16)
Thecompact finite difference method with Grünwald-Letnikovformula for Eq. (1),
(2) and (3) is in the form
 n (α ) n−k
n−k
n−k
n
n
n
α
n
n
n
∑ωk (ui −1 +10ui + ui+1 ) = 12S (ui −1 − 2ui + ui +1 ) +τ ( fi −1 + 10 fi + fi +1 )
k =0
i = 1,2,..., M −1, ....n = 1,2,..., N ,

u0 = 0, ....i = 1,2,..., M −1,
 i
u0n = f1 (tn ), ....uMn = f2 (tn ), ....n = 1,2,..., N.
α
Where S = τh 2 .
(17)
Compact Finite Difference Methods for the...
109
We can represent the equations defined in Eq. (17) in a linear system of equations
of the form
n
AU n = BU n −1 + ∑ CU n − k + τ α F n , 1 ≤ i ≤ M − 1 , n = 2, 3,...N
k =2
Where U n = [u1n , u2n ,....., uMn −1 ]T .
The matrices in Eq. (18) are tridiagonal and are defined as follows
 −(24S +10) 12S −1



 12S −1 −(24S +10) 12S −1


,
A=
⋱
⋱
⋱


12S −1 −(24S +10) 12S −1 


12S −1 −(24S +10) (M−1)×(M−1)

10 1

10 1





 1 10 1

 1 10 1



, B = −α 
, C = ωk(α ) 
⋱ ⋱ ⋱
⋱ ⋱ ⋱




1 10 1 
1 10 1 




1 10 (M −1)×( M −1)
1 10(M −1)×(M −1)


, k = 2, 3,..., n , 0 < α < 1
and the vectors F n for 2 ≤ n ≤ N are in the form
n
 −α

n −1
n
(α ) n − k
n
n
n
τ
(
α
u
+
(12
S
−
1)
u
−
ω
u
)
+
f
+
10
f
+
f
∑
k
0
0
0
0
1
2


k =2




f1n + 10 f 2n + f3n


Fn = 
⋮



f Mn −3 + 10 f Mn − 2 + f Mn −1


n
τ −α (α u n −1 + (12S − 1)u n − ω (α )u n −k ) + f n + 10 f n + f n 
∑
M
M
k
M
M −2
M −1
M 

k =2


(18)
110
Faoziya S. Al-Shibani et al.
4
Compact Finite Difference Method with RightShifted Grünwald-Letnikov Formula
To estimate the left-handed time fractional derivative in Eq. (1), we use the rightshifted Grünwald-Letnikovformula which is defined as( see [9] ).
∂ α u ( xi , t n )
1
= α
α
∂t
τ
n +1
∑= ω α u
( )
k
k
n − k +1
i
+ O (τ )
(19)
0
where p = 1.
This formula is shifted by one grid point to the right to get the fractional forward
difference formula.
We use finite difference approximation in Eq. (15) to discretize the second order
spatial derivative in Eq. (1) to obtain
n

∂ +  ∂ −u 


 1 n +1
 α ∑ ωk(α )uin − k +1 = ∂x 2 ∂x i + f i n ,
 h ∂+ ∂− 
τ k = 0
 1 + 12 ∂x ∂x 




i = 1, 2,..., M − 1, ....n = 1, 2,..., N ,
 0
ui = 0, ....i = 1, 2,..., M − 1,
 n
n
u0 = f1 (tn ), ....u M = f 2 (tn ), ....n = 1, 2,..., N .
(20)
Hence the compact finite difference method with right-shifted GrünwaldLetnikovformulafor Eq. (1), (2) and (3) is given as:
 n+1 (α ) n−k +1
n−k +1
+ uin+−1k +1 ) = 12S (uin−1 − 2uin + uin+1) +τ α ( fi−n1 +10 fi n + fi+n1)
∑ωk (ui−1 +10ui
k =0
i = 1,2,..., M −1,....n = 1,2,..., N,

u0 = 0, ....i = 1,2,..., M −1,
i
u0n = f1 (tn ),....uMn = f2 (tn ),....n = 1, 2,..., N.
(21)
The linear system of Eq. (4.3) is
n +1
AU n +1 = BU n − ∑ CU n − k +1 + τ α F n , 1 ≤ i ≤ M − 1 , 1 ≤ n ≤ N
k =2
(22)
Compact Finite Difference Methods for the...
111
where U n = [u1n , u2n ,....., uMn −1 ]T and the vector F n isdefined as
Fn =[(a + f0n +10 f1n + f2n ),( f1n +10 f2n + f3n ),.....,( fMn −3 +10 fMn −2 + fMn −1),(b + fMn −2 +10 fMn −1 + fMn )]T
n +1
where a = τ −α ((α + 12 S )u0n − u0n +1 − ∑ ωk(α )u0n − k +1 ) and
k =2
n +1
b = τ −α ((α + 12 S )uMn − uMn +1 − ∑ ωk(α )uMn − k +1 ) for 1 ≤ n ≤ N .
k =2
The entries ai , j , bi , j , ci , j and d i , j for i = 1, 2,...M − 1 and j = 1, 2,...M − 1 can be
defined as
α +12S....when...i = j −1..and..i = j +1

bi , j = 10α − 24S...when...i = j
0....otherwise

1....when...i = j −1..and..i = j +1

ai, j = 10...when...i = j
,
0....otherwise

ωk(α ) ....when...i = j −1..and..i = j +1

ci, j = 10ωk(α ) ...when...i = j
0....otherwise

5
, k = 2, 3,...n + 1 , 0 < α < 1
The Truncation Error
Let us consider the truncation error of Eq. (16). We have
∂+  ∂−u 
 
∂x  ∂x i
n
T ( xi , tn ) =
=
=
1
τ
1
τ
α
n
ωα u
∑
=
( ) n −k
k
i
−
k 0
 h ∂+ ∂− 
1 + 12 ∂x ∂x 


2
− fi n
∂ +  ∂ −u 
n
τα
k =0
n
ω
∑
=
k 0
n
 
 ∂α u   ∂2u 
∂x  ∂x i
+
−
 2
2
α 
 ∂t i  ∂x i 1 + h ∂+ ∂− 
 12 ∂x ∂x 


n
n
ω(α )uin−k − 
α ∑ k
1
(23)
(α ) n−k
k
i
u
n
n

 ∂α u   h4  ∂6u 

− α  +−
+
...
 6

 ∂t i  240  ∂x i

112
Faoziya S. Al-Shibani et al.
= O(τ p ) + O(h 4 )
Since we use the generating function (1 − z )α then the Grünwald-Letnikovformula
has an accuracy of order p = 1.
Hence
T ( xi , tn ) = O (τ ) + O ( h 4 )
(24)
The same result will be obtained if the truncation error of Eq. (20) is investigated.
Therefore both of the propsed fractional compact finite difference methodsfor the
time fractional diffusion equation have accuracy of order O(τ + h 4 )
6
The Stability Analysis of the Two Compact Finite
Difference Methods
In this section the von Neumann (or Fourier) method will be used to analyze the
stability of CFD methodwith the Grünwald-Letnikovformula and CFDmethod
with the right-shifted Grünwald-Letnikovformula.
We start with Eq. (17) and for simplicity let us write this formula with no source.
(α )
Cui [8] introduced the following lemma about properties of ωk
(α )
Lemma: The coefficients ωk ( k = 0,1,..) satisfy (see [8] )
(1) ω0(α ) = 1; ω1(α ) = −α ; ωk(α ) < 0, k = 1, 2,...
∞
(2)
∑ω α
k =0
(
k
)
m
= 0; ∀m ∈ N + , −∑ ωk(α ) < 1
k =1
Proposition 6.1: The compact finite difference scheme with the GrünwaldLetnikov formula defined in Eq. (17) is unconditionally stable.
Proof: Suppose U in is the approximate solution of Eq. (17) and so the error we
obtain from the difference between the theoretical and numerical solutions can be
addressed as
ε in = uin − U in ,
i = 0,1,..., M ,
n = 0,1,..., N .
This error can be presented by the same compact finite differencemethod with the
Grünwald-Letnikovformula.
Compact Finite Difference Methods for the...
113
We obtain
n
∑= ω α (ε
( )
k
+ 10ε in −k + ε in+−1k ) = 12 S (ε in−1 − 2ε in + ε in+1 ) , i = 1, 2,..., M − 1 ,
n−k
i −1
k 0
n = 1, 2,..., N .
(25)
ε 0n = ε mn = 0 , n = 1, 2,..., N
(26)
According to the Fourier method, the errors ε n ( xi ) at the grid points in the range
x ∈ [0,1] and a given time level can be expressed in terms of a finite Fourier series
with the complex exponential form( see [4] )
M
ε n ( xi ) = ∑ Am e
m =0
−1qm xi
, i = 0,1,..., M
(27)
where qm = mπ / l , l = Mh
To study the propagation of errors as t increases, let us omit the summation and
the constant Am, taking only a single term e
−1qm xi
.
Suppose the solution of equations (25) and (26) in the form
ε in = e β nτ e
−1qih
(28)
where e β nτ is termed the temporal or amplification factor and β is complex
temporal number which depends upon q
We note that at n = 0, the solution of error reduces to e
Neumann method, a scheme is said to be stable if and only if e
−1qih
βτ
. For the von
≤ 1 for all γ .
Substituting Eq.(28) into Eq.(25) we obtain
n
∑ ωkα e − k βτ =
k =0
−12 S sin 2 ( qh2 )
3 − sin 2 ( qh2 )
(29)
To study the stability of Eq.(29) let us consider the iteration when n → ∞; so we
obtain
114
Faoziya S. Al-Shibani et al.
n
lim ∑ ω k e
n →∞
α
−12 S sin 2 ( qh2 )
=
3 − sin 2 ( qh2 )
− k βτ
k =0
∞
∑ ωkα e − k βτ =
Hence
k =0
−12 S sin 2 ( qh2 )
3 − sin 2 ( qh2 )
(30)
(31)
Then from Eq.(8) we can obtain
(1 − e )
− βτ
α
−12 S sin 2 ( qh2 )
=
3 − sin 2 ( qh2 )
(32)
Since the maximum value of sin( qh ) = 1 , then
e βτ =
1
(33)
1 + ( 6 S )α
1
Therefore according to the von Neuman method the condition for stability
e β τ ≤ 1 is satisfied by
1
1 + ( 6S ) α
1
≤ 1 for all values ofS> 0 and 0 < α < 1 . Hence
the compact finite difference method with the Grünwald-Letnikovformula (17) is
τα
unconditionally stable for S = 2 > 0 and 0 < α < 1 . Now we study the stability
h
of the compact finite difference method with the right-shifted GrünwaldLetnikovformula.
Proposition 6.2: The compact finite difference scheme with the right2 α −1
shiftedGrünwald-Letnikovformula defined in Eq.(21) is stable if S ≤
for
3
0 <α <1
Proof: The error ε in satisfies the equations (21). This gives
n +1
∑ ω α (ε
k =0
( )
k
n − k +1
i −1
+ 10ε in − k +1 + ε in+−1k +1 ) = 12 S ( ε in−1 − 2ε in + ε in+1 ) , n = 0,1,..., N .
(34)
ε 0n = ε Mn = 0 , n = 0,1,..., N
n
β nτ
We assume that the solution of the equation (34) takes the form ε i = e e
(35)
−1qih
.
Compact Finite Difference Methods for the...
115
Inserting the above solution into Eq.(34), we obtain
e
βτ
−4 S sin 2 ( qh2 )
=
 1 2 qh  n +1 (α ) − k βτ
 1 − sin ( 2 )  ∑ ω k e
 3
 k =0
(36)
If e β τ > 1 for all β then the errors will propagate exponentially and the equation
of the error will be unstable
Suppose that e β τ ≤ 1 then − 1 ≤ e β τ ≤ 1
when e β τ ≤ 1 then from Eq.(36) we have
− 4 S sin 2 ( q2h )
≤1
n +1
1

qh 
2
(α ) β τ
 1 − sin ( 2 )  ∑ ω k e
3

 k =0
(37)
taking the maximume value of e β τ = 1 we get
− 4 S sin 2 ( q2h )
≤1
n +1
1

qh 
2
(α )
 1 − sin ( 2 )  ∑ ω k
3

 k =0
m
From the Lemma we have
∑ω α
k =0
( )
k
> −1 ∀ m ∈ N
+
then
S sin 2 ( qh2 ) > 0 is always true if S > 0
when e β τ ≥ − 1 we have
− 4 S sin 2 ( q2h )
≥ −1
n +1
1

qh 
2
(α ) − k β τ
 1 − sin ( 2 )  ∑ ω k e
3

 k −0
Taking the limits of the summation when n→∞ we obtain
4 S sin 2 ( q2h )
≥1
∞
1

qh 
2
(α ) − k β τ
 1 − sin ( 2 )  ∑ ω k e
3

 k −0
then from equation (8) we have
116
Faoziya S. Al-Shibani et al.
α
 1

4S sin 2 ( qh2 ) ≤ 1 − sin 2 ( qh2 )  (1 − e − βτ )
 3

Consider the extreme value e β τ = − 1 then
 1

4 S sin 2 ( qh2 ) ≤ 2α  1 − sin 2 ( qh2 ) 
 3

Since the maximum value of sin ( qh2 ) = 1 , then we arrive at the condition that
S ≤
7
2 α −1
.
3
Numerical Experiments
Let us consider the equation from Takaci et al. [3]
∂α u ( x, t ) ∂ 2u ( x, t ) 2e xt 2−α
=
+
− t 2 e x for
2
α
∂t
∂x
Γ(3 − α )
= 0.5
(38)
With the initial condition
( , 0) = 0,0 <
< 1
And the boundary conditions
(0, ) =
,
( , )=
C
(1, ) = C , 0 ≤
The exact solution of Eq. (38) is
(39)
≤
(40)
(41)
We implement the two numerical methods discussed in this paper for solving the
above example and compare their solutions with the exact solution.Figure 1 shows
the comparison of u(x,t) values between the solutions of the two methodsand the
exact solutionfor a certain set of parameter values which satisfy the stability
condition of the CFD scheme with the right-shifted Grünwald-Letnikovformula.
Compact Finite Difference Methods for the...
117
Fig.1: Comparison between the results of the CFD method with G-L formula and
CFD method with right-shifted G-L formulaand the exact solution at τ = 5.55556
× 10-10 and h = 0.1
To test the order of convergence of our schemes firstwe estimate the error in ∞
norm where e l = max uiN − U iN at = 1, ℎ = D = 0.2 and then decrease the
mesh size of ℎ to half and D to 0.125D. The results of the maximum errors and the
order of convergence of the compactDu Fort Frankel method are listed in table 1.
∞
1≤ i ≤ M −1
The order of convergence r (τ , h) is evaluated by the formula
(
r (τ , h) = log 2 e(8τ , 2h)
l∞
e(τ , h)
l∞
)
Table 2 represents the maximum errors and the order of convergence obtained by
the CFD method with the right-shifted Grünwald-Letnikovformula, here the step
sizesD, ℎ are shosen to satisfy the stability condition of this method.
Assuming that e l = C1τ + C2 h 4 where C1 and C2 are constant, the maximum error
∞
convergence tends to 4. Analogously, if D is large then e l ∞ ≈ C1τ and the order of
the convergence tends to 1 (Hu and Zhang, 2012).
4
tends to C2 h as time step size becomes small enough and the order of the
Table 1: Error in
∞
norm and order of convergence of CF with G-L method at
= 0.5
‖C‖F∞ order
h = τ = 0.2.
h = 0.1,τ = 0.025.
h = 0.05,τ = 3.125 E − 3
h = τ = 0.1
2.1081 E − 2
1.8383 E − 3
1.0299 E − 4
1.0713 E − 2
3.5195
4.1578
-
118
Faoziya S. Al-Shibani et al.
h = 0.05,τ = 0.0125
h = 0.025,τ = 1.5625 E − 3
Table 2: Error in
∞
8.1616 E − 4
3.7556 E − 5
3.7143
4.4418
norm and order of convergence of CF with shifted GLmethod at = 0.5
‖C‖F∞ order
h = 0.2,τ = 1.3889 E − 3
h = 0.1,τ = 1.7361E − 11
h = 0.05,τ = 2.17014 E − 12
h = 0.1,τ = 5.4254 E − 13.
h = 0.05,τ = 6.7817 E − 14
h = 0.025,τ = 8.4771E − 15
3.3687 E − 1
1.1923 E − 2
7.7921 E − 4
1.1656 E − 5
7.6203 E − 7
2.4355 E − 8
4.8204
3.9356
3.9351
4.9675
These results indicate the convergence of the proposedmethods.
8
Conclusion
In this paper, two high order compact finite difference schemes have been
developed. The stability of the proposod methods was investigated and it was
shown that the compact finite difference scheme with the GrünwaldLetnikovformulais unconditionally stable while the compact finite difference
scheme with the right-shifted Grünwald-Letnikovformula is conditionally stable.
Both of these methods have an accuracy of order >(ℎ? ) in space and
oforder>(D)in the temporal step. It should be noted that the proposed algorithms
produced reasonable results.
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