© 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.
IRE 1701500
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 .
ICONIC RESEARCH AND ENGINEERING JOURNALS
(3)
220