CONVERGENCE FOR NON-STATIONARY ADVECTION- DIFFUSION EQUATION

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© JUL 2019 | IRE Journals | Volume 3 Issue 1 | ISSN: 2456-8880

Convergence for Non-Stationary Advection- Diffusion Equation 1, 2

KHAING KHAING SOE WAI1, SAN SAN TINT2 Department of Engineering Mathematics, Technological University, Myitkyina, Myanmar

Abstract- In this paper, we first consider parabolic advection-diffusion problem. And, we study a characteristic Galerkin method for non- stationary advection diffusion equation. Then, we prove the stability and convergency of these method.

We assume that there exist two positive constants  0

Indexed Terms- Advection- Diffusion Equation, convergency, characteristic Galerkin method.

The parabolic advection-diffusion problem (1) can be reformulated in a weak form as follows:

I.

and

1 such that

1 div b(x)  a 0 (x)  1 2 For almost every x  . 0   0  (x) 

Given

INTRODUCTION

f  L2 (QT )

and

u 0  L2 (),

find

u  L (0, T; V) I C ([0, T]; L ()) such that 2

We assume that  is a bounded domain in R n=2, 3….with Lipschitz boundary and consider the parabolic initial boundary value problem: For each t  [0, T], we can find u (t) such that n,

0

2

d  u(t), v   a  u(t), v    f (t), v  , v  V  (3) dt   u(0)  u 0 ,  Where V  H10 () .

u   Lu  f in QT  (0,T)    t  u  0 on   (0,T)  ,  (1) T   u  u 0 on , for t  0,  Where L is the second-order elliptic operator

Lw   w 

n

 D (b w)  a w.(2) i

i 1

i

0

We consider the case in which  =

b

L (  )

.

n

i, j1

i 1

i

i

0

L (  )

 1.

IRE 1701398

initial datum u 0 . II.

A CHARACTERISTIC GALERKIN METHOD

a ij are smaller

bi ,i, j  1,..., n. without

loss of generality, we suppose that b is normalized to

b

u 0,h  Vh is an approximation of the

n

 D (b z)  a z,

whenever the diffusion coefficients than the adventive ones

d  (u h (t), v h )  a(u h (t), v h )  (f (t), v h ) ,  dt   v h  Vh , t  (0, T)  (4)  u h (0)  u 0,h .   1 Here Vh  H0 () is a suitable finite-dimensional space and

The "simplified" form considered in (2) is a perfect alias of the most general situation in which L is given by Lz    D (a D z)  i ij j

We write the semi-discrete (continuous in time) approximation of the advection-diffusion initial boundary value problem (3)

We define the characteristic lines associated to a vector field b  b(t, x). Being given x   and

s [0, T],

they

are

the

vector

ICONIC RESEARCH AND ENGINEERING JOURNALS

functions

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