© 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, j1
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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