© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880
The Semi-Implicit Scheme and Its Convergence for The Parabolic Problem 1,2
KHAING KHAING SOE WAI1, SAN SAN TINT2 Department of Engineering Mathematics, Technological University, Myanmar
Abstract -- In this paper, we consider the parabolic initial boundary value problem. Then, we describe the bilinear form of the parabolic advection- diffusion problem. . Finally, we prove the convergence of the semi implicit scheme for the problem. Indexed Terms - Advection- Diffusion Equation, semi implicit scheme, convergence
I.
By
using
Hölder's
inequality,
a (w, v)
w
v 0
0
1 C 2 v 2
0
bilinear
form
is
1 a(w, v) w v bi wDi v bi vDi w wv dx. 2 i 1 (2)
a , is continuous as follows:
a(w, v)
1 b 2
L ( )
n
(b w D v b v D w) dx
i
i 1
1 2
b
i
i
L ( )
i
w v dx
v w dx 1 w v dx.
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1
1
0
2
1
v 1 , where 11 .
t
n
v
Ch 2 . 1 2 2
(2.5)
For a small this becomes
The
w v dx
0
where C1C2 1 .
t
where V H10 () .
0
The stability condition for the -scheme is,
d u(t), v a u(t), v f (t), v , v V (1) dt u(0) u 0 ,
1 2
Poincaré's
Therefore the bilinear form a( , ) is continuous.
find u L2 0, T; V C0 0, T ; L2 such that
w vdx
v
w 0 1C1C2 w
a (w, v) w
Given f L2 (QT ) and u 0 L2 (),
0
2
We consider the parabolic advection-diffusion problem as follows:
We will prove that
1 C1 w 2
and
a (w, v) ( 1 1 ) ( w 0 ) 2 ( v 0 ) 2 ,
INTRODUCTION
inequality
w v dx
II.
Ch 2 . 1 2 2 1 12
NUMERICAL SCHEMES
We consider the semi-discrete (continuous in time) approximation of the advection-diffusion initial boundary value problem (1): d u h t , vh a u h t , vh f t , vh , dt v h Vh , t 0, T u h 0 u 0,h .
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220
© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 Here Vh H10 is a suitable finite-dimensional space and
u 0,h Vh approximates the initial datum
2u L2 0, T; L2 . Then t 2
u nh defined in (4) satisfies
for each n 0,1,..., N
u0. k u nh u t n I 1,h u t n exp C t n
2.1) A Semi -Implicit Scheme
0
We consider the semi-implicit time-discretization approach. This consists in evaluating the principal part of operator L at time level
t n 1 , whereas the
remaining parts are considered at the time tn. This scheme reads for
0
2 tn 2 u k k u 0,h 1,h u 0 C I 1,h s ds 0 0 t 0
C t
2
tn 0
2 u 2u s 2 s t t 1
n 0,1,..., N 1.
each
1
2 ds 0 2
(5)
where 1,h is the elliptic projection operator and C 0 is 1 a non-decreasing function of . a 0 u hn , v h f t n 1 , v h , v h Vh , (4) u 0h u 0,h, Proof k
1 n 1 n u h u h , vh u nh 1, vh bi u hn , Di vh t i 1 n
k Set nh u nh 1,h u t n in (4).
We study semi-discrete methods based on the socalled -scheme:
Then, we obtain
1 n 1 n u h u h , vh a u nh 1 1 u nh , vh f t n 1 1 f t n , vh , vh Vh , t u 0h u 0,h ,
1 n 1 k h 1,h u t n 1 hn 1,hk u t n , vh t
for each n 0, 1, ..., N 1, where 0 1, t T is the N
time step, N is a positive integer, and u 0,h Vh is a suitable approximation of the initial datum u0.
1. 2 k The elliptic projection operator 1,h is defined as
k 1,h
vV
v Vh
implies a
k 1,h
v , vh a v, vh
k bi hn bi 1,h u(t n ) , Di v h n
i 1
This includes the schemes: forward Euler 0 , backward Euler 1 and Crank-Nicolson
follows: For each
k a 0hn a 0 1,h u t n , vh f t n1 , vh .
That is, n 1 n 1 h nh , vh hn 1 , vh bi hn , Di vh a 0hn , vh t i 1 1 k f t n 1 , vh 1,h u t n 1 u t n , vh t
k 1,h u t n 1 , vh
vh Vh .
k nh 1 1,h u t n 1 , vh
n
k bi 1,h u t n , Di vh a 0 1,hk u t n , vh , i 1
2.2 Theorem Assume that u 0 H10 and that the solution u to (1) is
such
that
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u L2 0, T; H10 and t
1 n 1 h nh , vh nh 1, vh t bi hn , Di vh a 0hn , vh n
i 1
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© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 u 1 k t n 1 , v h 1,h u t n 1 u t n , v h t t
f t , v 1t u t u t , v u t , v
n 1
k 1,h
h
n 1
n
h
bi u t n , Di vh a 0 u t n , vh u t n , vh a u t , v u t , v i 1 u t , v u t , v b u t , D v b u t , D v a u t , v , n 1
n
k bi 1,h u t n , Di v h
h
i 1
n
k 1,h
0
k 1,h
n
k 1,h
n
h
i 1
i
n
n 1
i
h
k 1,h
h
n
n
h
k 1,h
n 1
h
n
k 1,h
i
i 1
n 1
i
h
0
h
1 n 1 h hn , vh hn1, vh t
(6)
bi hn , Di v h a 0nh , v h nh , v h , (7) n
On the other hand, the solution u satisfies
i 1
where h Vh is defined by the relation
u t n 1 , v h f t n 1 v h u t n 1 , v h t
n
u t u t , V ut t t , v
n
bi u t n 1 , Di v h a 0 u t n 1 , v h .
n 1
n h
n
n 1
h
1 t n 1 k I 1,h ut s ds, vh t t n
i 1
Thus, ( 6 ) becomes
h
n 1 n 1 h nh , vh hn 1 , vh bi hn , Di vh a 0hn , vh t i 1
n (8) u 1 k t n 1 , vh 1,h u t n 1 u t n , vh bi u t n , Di vh t t i 1
Therefore h satisfies a scheme like ( 4 ). we have n
k a 0 u t n , vh u t n , vh 1,h u t n , vh
n
k bi 1,h u t n 1 , Di vh a 0 1,hk u t n 1 , vh . i 1
Let us further recall that the definition of
reads k 1,h
n
u t n ), vh bi u t n , Di vh a 0 u t n , vh i 1
k u 0,h 1,h u0 0 2 t exp C1 n 1 a 0 L nh
2
2 0
Cl m 1 n t h n 0
2 1,h
(9)
.
The first term at the right hand side of (8) can be estimated as follows: u t n 1 u t n u 1 t n1 2u , v h s t n 2 s ds, v h t n 1 tn t t t t 2u 1 t s t n 2 s , v h ds t t t n 1
n
k 1,h u t n , vh bi 1,hk u t n , Di vh a 0 1,hk u t n , vh .
Since
i 1
k commutes with 1,h ( 6) can be rewritten t
n
1 t n1 2u s t n 2 s t t n t
v h 1 ds. 1,h
as n The second term at the right hand side of (8 ) can be 1 n 1 h hn , vh hn 1 , vh binh , Di v h a 0nh , v h estimated as follows: t i 1
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© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 1 t n1 u u 1 t n1 k k t n I 1,h s ds, vh t n I 1,h s t t t t
1,h
vh 1 ds.
b u t u t , D v n
k 1,h
i
i 1
n 1
n
i
k 1,h u t n u t n 1
h
k 1,h u t n u t n 1
vh
0
1,h
vh
2
1 t n1 u k t I 1,h s t t n
2
Moreover, we have
1
t t n1 2 u s 2 t 3 n t
n 2 h 1,h
2 ds 1,h 1
2 ds 1,h
1
2
k 1,h
u t u t n 1
n
1
0
a0
L ( )
k 1,h
u t u t n 1
n
,
0
and
a
0
u t u t , v u t u t v k 1,h u t n u t n 1 0 vh L ()
k 1,h
a0
a0
n 1
n
k 1,h
L ( )
h
n 1
n
h 1
1,h
1
.
h
u s t 2
n
tn
1 t n1 u k I 1,h s t t n t
k 1,h u t n 1 u t n
, v n h
h
vh
1
0
1,h
vh 1 a 0
0
vh 1 ds
k 1,h u t n 1 u t n
dx 1,h
vh 1 ,
0
1 t n1 I 1,hk ut s t t n
L ( )
1 a0
ds
nh
1,h
k 1,h u t n 1 u t n .
2 1,h
0
2 L ( )
t
t n 1 tn
2 L
t
t n 1 tn
By using inequality, we have
sup
v h Vh vh 0
, v vh
h
1
1 t
t n 1 tn
s t n
2
ds
1 2
t n1 2 u s t n t 2
1 t
ds t n 1
tn
t n 1 tn
I ut s
k 1,h u t n 1 u t n
a0
L ( )
2 L
k 1,h
0
k 1,h u t n 1 u t n ,
1
ds 1,h 2
1 2
k 1,h
2 0
,
2 u (s) ds , t 0
since the operator
2 u s ds , t 1
k is uniformly bounded in
2 0
2 m 1 2 CC 1 t n1 u k k u 0,h 1,h u 0 0 1 2 t tn I 1,h s ds t n 0 t 0
C3 t
t n 1 tn
2 u 2u s 2 s t t 1
t ds exp C1 n 1 a 0 0
2
2 L
2 2 2 k u nh u t n I 1,h u t n 0 exp C t n u0,h 1,hk u0 0 0
0
C
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k 1,h u t n 1 u t n
ds 1,h
1,h 2 ds H1 . From (7), we obtain 0 1,h 2
nh 1 2
2
0
2
2 2 2 t n 1 u 1 t n1 u k C2 t s ds I s 1,h t ds 2 t t n t n t 0 0
1 a0
1,h
1,h
1 t n1 I 1,hk ut s t t n
1,h
vh 1 ds
L ( )
k 1,h u t n 1 u t n a 0
nh
ds
+
1 t n1 2u s t n 2 s t t n t
n h
2
2 2 2 t n 1 u 1 t n1 u k C2 t s ds I s ds 1,h 2 t t n t t n t 0 0
2
t n 1
n h
2 t n 1 u C2 t s 2 t n t
k 1,h u t n 1 u t n a 0
Therefore, (8) becomes
, v 1t s t
n 2 h 1,h
tn 0
I ut s ds 0 2
k 1,h
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,
© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 2 u 2u C t s 2 s 0 t t 1 k where nh u nh 1,h u t n .
2
tn
2
0
ds
,
k u nh u t n I 1,h u t n exp C t n 0
C
tn 0
C t
0
I 1,hk 2
tn 0
u
0,h
k 1,h u0
2 0
u s ds t 0 2
2 u 2u s 2 s t t 1
III.
1
2 ds . 0 2
CONCLUSION
This paper has presented parabolic advectiondiffusion problem. Then, we study numerical schemes for initial boundary value problem. And then, we prove convergence of the semi implicit problem.
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