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1. International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISSN: 2319-6491
Volume 2, Issue 11 (July 2013) PP: 01-08
www.ijeijournal.com Page | 1
The Finite Difference Methods And Its Stability For Glycolysis
Model In Two Dimensions
Fadhil H. Easif1
, Saad A. Manaa2
1,2
Department of Mathematics, Faculty of Science, University of Zakho, Duhok, Kurdistan Region, Iraq
Abstract: The Glycolysis Model Has Been Solved Numerically In Two Dimensions By Using Two Finite
Differences Methods: Alternating Direction Explicit And Alternating Direction Implicit Methods (ADE And
ADI) And We Were Found That The ADE Method Is Simpler While The ADI Method Is More Accurate. Also,
We Found That ADE Method Is Conditionally Stable While ADI Method Is Unconditionally Stable.
Keywords: Glycolysis Model, ADE Method, ADI Method.
I. INTRODUCTION
Chemical reactions are modeled by non-linear partial differential equations (PDEs) exhibiting
travelling wave solutions. These oscillations occur due to feedback in the system either chemical feedback (such
as autocatalysis) or temperature feedback due to a non-isothermal reaction.
Reaction-diffusion (RD) systems arise frequently in the study of chemical and biological phenomena
and are naturally modeled by parabolic partial differential equations (PDEs). The dynamics of RD systems has
been the subject of intense research activity over the past decades. The reason is that RD system exhibit very
rich dynamic behavior including periodic and quasi-periodic solutions and chaos(see, for example [8].
1.1. MATHEMATICAL MODEL:
A general class of nonlinear-diffusion system is in the form
𝜕𝑢
𝜕𝑡
= 𝑑1∆𝑢 + 𝑎1 𝑢 + 𝑏1 𝑣 + 𝑓 𝑢, 𝑣 + 𝑔1 𝑥
𝜕𝑣
𝜕𝑡
= 𝑑2∆𝑢 + 𝑎2 𝑢 + 𝑏2 𝑣 − 𝑓 𝑢, 𝑣 + 𝑔2 𝑥
(1)
with homogenous Dirchlet or Neumann boundary condition on a bounded domain Ω , n≤3, with locally
Lipschitz continuous boundary. It is well known that reaction and diffusion of chemical or biochemical species
can produce a variety of spatial patterns. This class of reaction diffusion systems includes some significant
pattern formation equations arising from the modeling of kinetics of chemical or biochemical reactions and from
the biological pattern formation theory.
In this group, the following four systems are typically important and serve as mathematical
models in physical chemistry and in biology:
 Brusselator model:
𝑎1 = − 𝑏 + 1 , 𝑏1 = 0, 𝑎2 = 𝑏, 𝑏2 = 0, 𝑓 = 𝑢2
𝑣, 𝑔1 = 𝑎, 𝑔2 = 0,
where a and b are positive constants.
 Gray-Scott model:
𝑎1 = − 𝑓 + 𝑘 , 𝑏1 = 0, 𝑎2 = 0, 𝑏2 = −𝐹, 𝑓 = 𝑢2
𝑣, 𝑔1 = 0, 𝑔2 = 𝐹,
where F and k are positive constants.
 Glycolysis model:
𝑎1 = −1, 𝑏1 = 𝑘, 𝑎2 = 0, 𝑏2 = −𝑘, 𝑓 = 𝑢2
𝑣, 𝑔1 = 𝜌, 𝑔2 = 𝛿,
 where 𝑘, 𝑝, and 𝛿 are positive constants
 Schnackenberg model:
𝑎1 = −𝑘, 𝑏1 = 𝑎2 = 𝑏2 = 0, 𝑓 = 𝑢2 𝑣, 𝑔1 = 𝑎, 𝑔2 = 𝑏,
where k, a and b are positive constants.
Then one obtains the following system of two nonlinearly coupled reaction-diffusion equations (the Glycolysis
model),
𝜕𝑢
𝜕𝑡
= 𝑑1∆𝑢 − 𝑢 + 𝑘𝑣 + 𝑢2 𝑣 + 𝜌, 𝑡, 𝑥 ∈ 0, ∞ × Ω
𝜕𝑣
𝜕𝑡
= 𝑑2∆𝑣 − 𝑘𝑣 − 𝑢2
𝑣 + 𝛿, 𝑡, 𝑥 ∈ 0, ∞ × Ω
𝑢 𝑡, 𝑥 = 𝑣 𝑡, 𝑥 = 0, 𝑡˃0, 𝑥𝜖𝜕Ω
𝑢 0, 𝑥 = 𝑢0 𝑥 , 𝑣 0, 𝑥 = 𝑣0 𝑥 , 𝑥𝜖Ω
(2)
where 𝑑1, 𝑑2, 𝑝, 𝑘 𝑎𝑛𝑑 𝛿 are positive constants [9].

2. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 2
II. MATERIALS AND METHODS
2.1. Derivation Of Alternating Direction Explicit (ADE) For Glycolysis Model:
The two dimensional Glycolysis model is given by
𝜕𝑢
𝜕𝑡
= 𝑑1
𝜕2
𝑢
𝜕𝑥2
+
𝜕2
𝑢
𝜕𝑦2
− 𝑢 + 𝐾𝑣 + 𝑢2
𝑣 + 𝜌, (𝑡, 𝑥) ∈ (0, ∞) × Ω
𝜕𝑣
𝜕𝑡
= 𝑑2
𝜕2
𝑣
𝜕𝑥2
+
𝜕2
𝑣
𝜕𝑦2
− 𝐾𝑣 − 𝑢2
𝑣 + 𝑑𝑒𝑙, (𝑡, 𝑥) ∈ (0, ∞) × Ω
(2)
We consider a square region 0≤x≤1, 0≤y≤1 and u, v are known at all points within and on the
boundary of the square region. We draw lines parallel to x, y, t–axis as x=ph, y=qk, and t=nz, p,q=0, 1, 2,…,M
and n=0, 1, 2,…, N, where h=δx, k= δy, z= δt.
Denote the values of u at these mesh points by nqpunzqkphu ,,),,(  and nqpvnzqkphv ,,),,(  . The
explicit finite difference representation of the Glycolysis model is
,,,
2
,,,,),1,,,2,1,,,1,,2,,1(
2
2,,1,,
,,
2
,,,,,,),1,,,2,1,,,1,,2,,1(
2
1,,1,,
delnqpvnqpunqpKvnqpvnqpvnqpvnqpvnqpvnqpv
h
d
z
nqpvnqpv
nqpvnqpunqpKvnqpunqpunqpunqpunqpunqpunqpu
h
d
z
nqpunqpu





and
delnqpvnqpzunqpzKvnqpvnqpvnqpvnqpvnqpvnqpv
h
zd
nqpvnqpv
znqpvnqpzunqpzKvnqpzunqpunqpunqpunqpunqpunqpu
h
zd
nqpunqpu


.,,
2
,,,,),1,,,2,1,,,1,,2,,1(
2
2
,,1,,
,,
2
,,,,,,),1,,,2,1,,,1,,2,,1(
2
1
,,1,, 
Assume that h=k and .2,1,
2
 i
h
zid
im Then simplifying the system to obtain
,,,
2
,,,,),1,,,2,1,,,1,,2,,1(2,,1,,
,,
2
,,,,,,),1,,,2,1,,,1,,2,,1(1,,1,,
delnqpvnqpzunqpzKvnqpvnqpvnqpvnqpvnqpvnqpvmnqpvnqpv
znqpvnqpzunqpzKvnqpzunqpunqpunqpunqpunqpunqpumnqpunqpu

 
and
delnqpvnqpzunqpzKvmnqpunqpunqpunqpvnqpvmnqpv
znqpvnqpzunqpzKvznqpzunqpunqpunqpunqpumnqpuBzmnqpu


.,,
2
,,,,2),1,,1,,,1,,1(,,)241(1,,
,,
2
,,,,,,),1,,1,,,1,,1(1,,))1(141(1,, 
This is the alternating direction explicit formula for the Glycolysis model
Derivation Of Alternating Direction Implicit (ADI)For Glycolysis Model:
In the ADI approach, the finite difference equations are written in terms of quantities at two x levels.
However, two different finite difference approximations are used alternately, one to advance the calculations
from the plane n to a plane n+1, and the second to advance the calculations from (n+1)-plane to the (n+2)-
plane by replacing
𝜕2 𝑢
𝜕𝑥2 and
𝜕2 𝑣
𝜕𝑥2 by implicit finite difference approximation [5]. we get
,,,
2
,,,,),1,,,2,1,(
2
2)1,,11,,21,,1(
2
2,,1,,
,,,
2
,,
,,,,),1,,,2,1,(
2
1)1,,11,,21,,1(
2
1,,1,,
delnqpvnqpunqpKvnqpvnqpvnqpv
K
d
nqpvnqpvnqpv
h
d
z
nqpvnqpv
nqpvnqpu
nqpKvnqpunqpunqpunqpu
k
d
nqpunqpunqpu
h
d
z
nqpunqpu






and
.,,
2
,,,,)2,1,2,,22,1,(
2
2
)1,,11,,21,,1(
2
21,,2,,
,,,
2
,,
,,,,)2,1,2,,22,1,(
2
1
)1,,11,,21,,1(
2
11,,2,,
delnqpvnqpunqpKvnqpvnqpvnqpv
K
d
nqpvnqpvnqpv
h
d
z
nqpvnqpv
nqpvnqpu
nqpKvnqpunqpunqpunqpu
k
d
nqpunqpunqpu
h
d
z
nqpunqpu







3. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 3
Simplifying the system will give
.1.,,,
2
,,,,),1,,,2,1,(
2
2
)1,,11,,21,,1(
2
2
1,,
,,
,,,,
2
,,,,),1,,,2,1,(
2
1
)1,,11,,21,,1(
2
1
1,,
zdelnqpzvnqpvnqpzunqpKvnqpvnqpvnqpv
K
zd
nqpvnqpvnqpv
h
zd
nqpv
nqpuz
nqpzKvnqpvnqpzunqpzunqpunqpunqpu
k
zd
nqpunqpunqpu
h
zd
nqpu




From the second two equations we will obtain a system of the form
.1,,,,
2
,,,,)2,1,2,,22,1,(
2
2)1,,11,,21,,1(
2
2
2,,
,,
,,,,
2
,,,,)2,1,2,,22,1,(
2
1)1,,11,,21,,1(
2
1
2,,
zdelnqpzvnqpvnqpzunqpzKvnqpvnqpvnqpv
k
zd
nqpvnqpvnqpv
h
zd
nqpv
nqpuz
nqpzKvnqpvnqpzunqpzunqpunqpunqpu
k
zd
nqpunqpunqpu
h
zd
nqpu




From the first two equations where ,
2
1
2,
2
1
1
k
zd
r
h
zd
r 
2
2
1
h
zd
m  and 2
2
2
k
zd
m  , we get
,.,,,
2
,,
,,),1,,,2,1,(2)1,,11,,21,,1(11,,
,,,,,
2
,,
,,,,),1,,,2,1,(2)1,,11,,21,,1(11,,
nqpvzdelnqpvnqpzu
nqpzKvnqpvnqpvnqpvmnqpvnqpvnqpvmnqpv
nqpuznqpvnqpzu
nqpzKvnqpzunqpunqpunqpurnqpunqpunqpurnqpu





and this implies that
zdelnqpvnqpzunqpvmnqpvnqpvmnqpvnqpvnqpvmnqpv
nqpzKvznqpvnqpzunqpuzrnqpunqpurnqpunqpunqpurnqpu


.,,
2
,,,,)221(),1,,1,(2)1,,11,,21,,1(11,,
,,,,
2
,,,,)121(),1,,1,(2)1,,11,,21,,1(11,, 
We have simplifying these systems of equations yields
,,,
2
,,
,,)221(),1,,1,(2)1,,11,,1(11,,)121(
,,
2
,,,,
,,)221(),1,,1,(2)1,,11,,1(11,,)121(
nqpvnqpzu
nqpvmnqpvnqpvmnqpvnqpvmnqpvm
nqpvnqpzunqpzKvz
nqpuzrnqpunqpurnqpunqpurnqpur




also the second system of equations, yields
.,,
2
,,
)2,1,2,1,(21,,)121()1,,11,,1(12,,)221(
,,,,
2
,,
,,)2,1,2,1,(21,,)121()1,,11,,1(12,,)221(
zdelnqpvnqpzu
nqpvnqpvmnqpvmnqpvnqpvmnqpvm
znqpzKvnqpvnqpzu
nqpzunqpunqpurnqpurnqpunqpurnqpur





The last two systems represent Alternating Direction implicit under the conditions
.01,1,1,,
01,1,11,,1


nqMunqMu
nqunqu
Also we have
.0
0
1,1,1,,
1,1,11,,1




nqMnqM
nqnq
vv
vv
The tridiagonal matrices for the system in the level n advanced to the level n+1, for both u and v can be
formulated as follows AU=B.



































)121(10000000
1)121(1000000
.........
.........
...00...
...00..
..100.
1)121(100
000001)121(10
0000001)121(1
0...0001)121(
rr
rrr
r
rrr
rrr
rrr
rr





































1,,1
1,2
1,3
.
.
.
1,5
1,,4
1,,3
1,,2
nqMu
qnu
qnu
qu
nqu
nqu
nqu
=

4. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 4










































nqMvnqMzunqMuzrnqMunqMurz
nqvnqzunquzrnqunqurz
nqvnqzunquzrnqunqurz
nqvnqzunquzrnqunqurz
,,1,,1
2
,,1)221(),1,1,1,1(2
.
.
.
.
,,4,,4
2
,,4)221(),1,4,1,4(2
,,3,,3
2
,,3)221(),1,3,1,3(2
,,2,,2
2
,,2)221(),1,2,1,2(2







































)121(10000000
1)121(1000000
.........
.........
...00...
...00..
..100.
1)121(100
000001)121(10
0000001)121(1
0...0001)121(
mm
mmm
m
mmm
mmm
mmm
mm





































1,,1
1,,2
1,,3
.
.
.
1,,5
1,,4
1,,3
1,,2
nqMv
nqMv
nqMv
nqv
nqv
nqv
nqv
=








































nqMvnqMzunqMvzKmnqMvnqMvm
nqvnqzunqvzKmnqvnqvm
nqvnqzunqvzKmnqvnqvm
nqvnqzunqvzKmnqvnqvm
,,1,,1
2
,,1)221(),1,1,1,1(2
.
.
.
.
,,4,,4
2
,,4)221(),1,4,1,4(2
,,3,,3
2
,,3)221(),1,3,1,3(2
,,2,,2
2
,,2)221(),1,2,1,2(2
And the tridiagonal matrices for the system in level n+1 advanced to level n+2 for both u and v are
given by AU=B.



































)221(20000000
2)221(2000000
.........
.........
...00...
...00..
..200.
2)221(200
000001)121(10
0000002)221(2
0...0002)221(
rr
rrr
r
rrr
rrr
rrr
rr





































2,1,
2,2,
2,3,
.
.
.
2,5,
2,4,
2,3,
2,2,
nMpu
nMpu
nMpu
npu
npu
npu
npu
=

5. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 5









































1,1,2,
2
,2,11,1,)121()1,1,11,1,1(1
.
.
.
.
,5,,5,
2
,5,1,5,)121()1,5,11,5,1(1
,4,,4,
2
,4,1,4,)121()1,4,11,4,1(1
,3,,3,
2
,3,1,3,)121()1,3,11,3,1(1
,2,,2,
2
,2,1,2,)121()1,2,11,2,1(1
npMvnpzunpMzunMpuzrnMpunMpurz
npvnpzunpzunpuzrnpunpurz
npvnpzunpzunpuzrnpunpurz
npvnpzunpzunpuzrnpunpurz
npvnpzunpzunpuzrnpunpurz





Also for AV=B the tridiagonal is in the form



































)221(20000000
2)221(2000000
.........
.........
...00...
...00..
..100.
2)221(200
000002)221(20
0000002)221(2
0...0002)221(
mm
mmm
m
mmm
mmm
mmm
mm





































2,1,
2,2,
2,3,
.
.
.
2,5,
2,4,
2,3,
2,2,
nMpv
nMpv
nMpv
npv
npv
npv
npv
=








































nMpvnMpzunMpzunMpvzKmnMpvnMpvm
npvnpzunpzunpvzKmnpvnpvm
npvnpzunpzunpvzKmnpvnpvm
npvnpzunpzunpvzKmnpvnpvm
,1,,1,
2
,1,1,1,)121()1,1,11,1,1(1
.
.
.
.
,4,,4,
2
,4,1,4,)121()1,4,11,4,1(1
,3,,3,
2
,3,1,3,)121()1,3,11,3,1(1
,2,,2,
2
,2,1,2,)121()1,2,11,2,1(1
2.2. Numerical Stability In Two Dimensional Spaces:
2.3.1 Numerical stability of ADE: The Von-Neumann method has been used to study the stability analysis of
Glycolysis model in two dimensions; we can apply this method by substituting the solution in finite difference
method at time t by 1and0,where,)(  m
ym
e
xm
et 

 . To apply Von-Neumann on the first
equation of Glycolysis model
,
2
]
2
2
2
2
[2
,
2
]
2
2
2
2
[1
delvuKv
y
v
x
v
d
t
v
vuKvu
y
u
x
u
d
t
u



















For the first equation after linearizing it and for some values of 𝜌 and K neglect the terms 𝜌𝑧 and
nqpzKv ,, [3], will be in the form

6. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 6
.)()()(
)(()()41()(
)()()(
)(
11
yymxmyymxmymxxm
ymxxmymxmymxm
eeteeteet
eetreetzreett








Now dividing both sides of the above equation by
ymxm
eet 
 )( to obtain
)).2/(sin21)2/(sin21()41(
)cos(2)cos(2()41(
)()41(
)(
)(
22
11
11
11
yxrzr
yxrzr
eeeerzr
t
tt ymymxmxm



 



 
For some values of  and , )2/(sin2
x and )2/(sin2
y are unity [4], so
.)81(4)41(
)2(2)41(
)(
)(
111
11






zrrzr
rzr
t
tt
For stable situation, we need 1||  , so 1)181(1  zr ,
Case 1: )181(1 zr 
8
2
1
z
r


Case 2: .01becausecasethisneglect
8
11)181( 

 r
z
rzr
For the second (Linearized) equation of Glycolysis model which is in the form
).,1,,1,,,1,,1(2,,)241(1,, nqpvnqpvnqpvnqpvrnqpvKzrnqpv 
Let ym
e
xm
etnqpV

 )(,,  and substituting it in the above equation to obtain
)
)(
)(
)(
)(
)(
)((2
)(
)()241()(
yym
e
xm
et
yym
e
xm
et
ym
e
xxm
etr
ym
e
xxm
etKzr
ym
e
xm
ett


















Dividing both sides of the above equation by ym
e
xm
et

 )( to obtain
).281(
)2/(
2
sin24)2/(
2
sin241(
)]2/(
2
sin21)2/(
2
sin21[22)241(
)]cos(2)cos(2[2)241(
][2)241(
)(
)(
Kzr
Kzyrxr
yxrKzr
yxrKzm
ym
e
ym
e
xm
e
xm
emKzr
t
tt




















For some values of  and , we can assume that )2/(
2
sin x and )2/(
2
sin y are unity [4], so
)281(
)(
)(
Kzr
t
tt





the equation is stable if 1 which implies that
1)281(11|Kz-281|  Kzrr , which are located in two cases
Case 1:
8
2
2228)281(1
Kz
rKzrKzr

 and
Case 2: 8/2281)281( KzrKzrKzr  .
So the system is stable under the conditions
8
2
1
z
r

 ,and
8
2
2
Kz
r

 .
2.3.2 Numerical stability of ADI: The ADI finite difference form for the first equation of Glycolysis model is
given in the form
znqpvnqpzunqpzKvnqpuzr
nqpunqpurnqpunqpurnqpur


,,
2
,,,,.,,)221(
),1,,1,(2)1,,11,,1(11,,)121(
By linearization the term 2
,, nqpzu vanishes, and for the same values of K and 𝜌 the terms nqpzKv ,, , z vanishes.
Then the equation take the form

7. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 7
.,,)221(
),1,,1,(2)1,,11,,1(11,,)121(
nqpuzr
nqpunqpurnqpunqpurnqpur


In order to study the stability of the above equation, after letting ym
e
xm
etnqpu

 )(,,  , where 1m , we
obtain
.)()121()
)()(
)(((2
)
)()(
)(((1)()121(
ym
e
xm
etzr
yym
e
xm
e
yym
e
xm
etr
ym
e
xxm
e
ym
e
xxm
ettr
ym
e
xm
ettr

















Dividing both sides of the above equation by ym
e
xm
e

to get
).()121(
)
)(((2)
)
)(((1)()121( tzr
ym
e
ym
etr
xm
e
xm
ettrttr 



 








Rearranging, we get
).())1(121()]2/(
2
sin21(2)[(2)]2/(
2
sin21(2)[(1
)()221()cos(2)((1)cos(2))((1
tbkrytrxttr
tzrytrxttr




For some values of  and , assume that )2/(
2
sin x and )2/(
2
sin y are unity [4] , so the equation will take
the form
.
)141(
)241(
)(
)(
)()141()()141(
)()221()2)((2)2)((1)()121(












r
zr
t
tt
tzrttr
tzrtrttrttr
So .1
141
)24(1
)(
)(








r
zr
t
tt
Similarly, for the second equation of the Glycolysis model we assume that ym
e
xm
etnqpv

 )(,,  , where
1m ,the finite difference form of this equation is
1,1,)221(
)1,,11,,1(2)1,,11,,1(11,,)121(


nqpvrKz
nqpvnqpvrnqpvnqpvrnqpvr
Then
.)()221()
)()(
)(((2
)
)()(
)((1)()221(
ym
e
xm
etrKz
yym
e
xm
e
yym
e
xm
etr
ym
e
xxm
e
ym
e
xxm
ettr
ym
e
xm
ettr

















Dividing both sides of the above equation by ym
e
xm
e

, to get
.)()
)(
)()(2
)(
)((2
)
)(
)(
)(
)((2)()221(
ym
e
xm
et
yym
e
xm
et
ym
e
xm
et
yym
e
xm
etr
ym
e
xxm
ett
ym
e
xxm
ettr
ym
e
xm
ettr























Which implies that
).()
2
21()()2/(
2
sin21(2(2)]2/(
2
sin21(2)[(1)()121(
)()
2
21()]cos(2)[(2)]cos(2)[(1)()121(
trKztyrxttrttr
trKzytrxttrttr




For some values of  and , we have 2/(
2
sin x ) and )2/(
2
sin y are unity, so we have
).()
2
21()(2)2)((1)()121( trKztrttrttr  
)()
2
41()()141( trKzttr  
.
2)141(
)24(1(
)(
)(








r
Kzr
t
tt
Where .
1
 and .
2
 stand for the I-plane and II-plane respectively, each of the above terms is conditionally
stable. However the combined two-levels has the form
].
)141(
))24(1(
][
)141(
))24(1(
[
21 r
Kzr
r
Kzr
ADI 



 

8. The Finite Difference Methods and Its Stability for Glycolysis Model in Two Dimensions
www.ijeijournal.com Page | 8
Thus the above scheme is unconditionally stable, each individual equation is conditionally stable by itself, and
combined two-level is completely stable.
III. APPLICATION (NUMERICAL EXAMPLE)
We solved the following example numerically to illustrate the efficiency of the presented methods,
suppose we have the system
𝜕𝑢
𝜕𝑡
= 𝑑1∆𝑢 − 𝑢 + 𝑘𝑣 + 𝑢2
𝑣 + 𝜌,
𝜕𝑣
𝜕𝑡
= 𝑑2∆𝑣 − 𝑘𝑣 − 𝑢2
𝑣 + 𝛿
We the initial conditions
U(x, 0) = Us + 0.01 sin(x/ L) f
or 0 ≤x ≤ L
V (x, 0) = Vs – 0.12 sin(x/ L) for 0 ≤x ≤ L
U(0, t) = Us , U(L, t) = Us and V (0, t) = Vs, V (L, t) = Vs
We will take
d1=d2=0.01 , 𝜌= 0.09 , 𝛿=-0.004 , Us=0 , Vs=1
Then the results in more details are shown in following table and figure:
IV. CONCLUSION
We saw that alternating direction implicit is more accurate than alternating direction explicit method
for solving Glycolysis model and we found that ADE method is stable under condition 𝑟1 ≤
2−𝑧
8
, and 𝑟1 ≤
2−𝐾𝑧
8
,
while ADI is unconditionally stable.
REFERENCES
[1]. Ames,W.F.(1992);’Numerical Methods for Partial Differential Equations’3rd
ed.Academic, Inc. Dynamics of PDE, Vol.4, No.2,
167-196, 2007.
[2]. Ellbeck, J.C.(1986)” The Pseude-Spectral Method and Path Following Reaction –Diffusion Bifurcation Studies”SIAM
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1.0003 1.0000
1.0017 1.0013
1.0028 1.0024
1.0036 1.0032
1.0040 1.0038
1.0042 1.0041
1.0042 1.0043
1.0041 1.0044
1.0040 1.0044
1.0038 1.0044
1.0038 1.0044
1.0038 1.0044
1.0040 1.0044
1.0041 1.0044
1.0042 1.0043
1.0042 1.0041
1.0040 1.0038
1.0036 1.0032
1.0028 1.0024
1.0017 1.0013
1.0003 1.0000