THE SEMI-IMPLICIT SCHEME AND ITS CONVERGENCE FOR THE PARABOLIC PROBLEM

Page 1

© 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    11 .

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


Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.