CONVERGENCE FOR NON-STATIONARY ADVECTION- DIFFUSION EQUATION

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

0

L (  )

 1.

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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

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280

dX (t;s, x)  b  t, X (t;s, x)  , dt X(s;s, x)  x.

 t (0, T)   (5)  

The existence and uniqueness of the characteristic lines for each choice of s and x hold under suitable assumptions on b, for instance b continuous in [0, T]   and Lipschitz continuous in  , uniformly

with respect to t [0, T]. From a geometric point of view, X(t;s, x) provides the position at time t of a particle which has been driven by the field b and that occupied the position x at the time s. The uniqueness result gives in particular that

X  t;s, X(s; , x)   X(t; , x)(6) For each t,s, [0, T] and x   .

X  t;s, X(s; t, x)   X(t; t, x)  x,

i.e., for fixed t and s, the inverse function of x  X(s; t, x) is given by y  X(t;s, y). we

define

u (t, y)  u  t, X(t;0, y)    (7) or equivalenty, u (t, x)  u  t, X(0; t, x)  . From (5), it follows that u u (t, y)   t, X(t;0, y)   t t

n

 Di u  t, X(t;0, y)  i 1

dXi (t;0, y) dt

u    b u   t, X(t;0, y)  .(8)  t  We can rewrite the non-stationary advectiondiffusion equation as

u   u  (div b  a0 ) u  f t

(9) InQ T

The time derivative is approximated by the backward Euler scheme,

u  t , y  u  tn , y u .(10)  t n 1, y   n 1 t t

If we set

X  t n ;0, y   X  t n ;0, X(0; t n 1 , x)   X  t n ; t n 1, x  . (3.32)

Then we obtain

u  t , x   u  t n , X(t n ; t n 1 , x)  u .  t n 1 , X(0; t n 1 , x)   n 1 t t

We

denote

X  t n ; t n 1 , x 

a

suitable

approximation

of

by

X (x), n  0,1,..., N  1, us n

can write the following implicit discretization scheme for problem ( ). If we set

u 0  u 0 , then for n  0,1,..., N  1 we

solve

u n 1  u n o X n    u n 1  div b  t n 1   a 0  u n 1  f  t n 1  t In . (11) A boundary condition has to be imposed on  . We (1) condition consider the homogeneous Dirichlet

u n 1 |  0.

Hence

Therefore,

u  t n , y   u  t n , X(t n ;0, y) 

that

y  X  0; t n 1 , x  , we have

u  t n 1 , y   u  t n 1 , X(0; t n 1 , x)   u  t n 1 , x  .

We choose a backward Euler scheme also for discretizing

dX  t; t n 1 , x   b  t, X(t; t n 1 , x)  .(12) dt



t n 1 tn

dX  t; t n 1 , x  dt  dt



t n 1 tn

b  t; X(t; t n 1 , x)  dt.

This produces the following approximation of

X  t n ; t n 1 , x  : n X(1) (x)  x  b  t n 1 , x  t.(13)

Here

n X(1) is a second order approximation of

X  t n ; t n 1 , x  , since we are integrating (12) on the time interval

 t n , t n 1  , which has length t .

A more accurate scheme is provided by the second order Runge-Kutta scheme,

t   n X(2) (x)  x  b  t n 1 2 , x  b (t n 1 , x)  t, (14) 2  (3.41) Which gives a third order approximation of X  t n ; t n 1 , x  . n It is necessary to verify that X(i) (x)  for

each x  , i  1, 2, so that we can compute

From (7), we have

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n From (15), it follows that the map X(1) is injective.

t [0, T] and x   .

Therefore, we introduce the change of variable

As

a

X (x)  x n (i)

consequence,

n n 1 y  X(1) (x), and setting Y(1)n (y)   X(1)  (y), we

for

have

* x   , i  1, 2. If we denote by x    the point

 u

having minimal distance from x  , we have

X (x)  x  b  t n 1 , x  t,

 t, bt , x  bt

x  

*

* n 1 , x 

n 1

xx

g Lip(  )

 1  tC1 Jac b  t n 1 

n u n o X (1)

n u n o X(1)

For each t [0, T] and almost every x , stability is proven. In fact, multiplying (11) by u and integrating over  , we obtain

n 1



n (1)



n 1

0



0

By using Hölder's inequality, we have u n 1

2 0

  t u n 1

2 0



n 1

n  u n o X (1)  t f  t n 1  0

n 1

0



(y)  1  C1 1*t  dy 1

2

0

for

we

finally

obtain

each

dx

dx.

u n 1 .

2

u n 1 u

n  u n o X(1)

0

n 1

0

 t f  t n 1  0 .

0

by using Poincarè's inequality, we have

u n 1



n 1 2

n

 1  tC31*  u n .(20)

u n 1  t

f  t n 1  u n 1 dx.

  u  dx   u o X  u dx  t  u u  t   div b (t )  a   u  dx  t  f  t  u n

0

u

n  0,1,..., N  1,

By using Gauss-divergence theorem, we have n 1 2

2

n X(1) ()

enough). From (17),



0





C2 is small

dx    u u dx



2

Therefore condition (19) implies (15) (if

n 1 n 1

2

and

L (  )

(18), we have

(16)

  div b  t n 1   a 0   u n 1  dx  

t[0,T]

C1  0, 0  C2  C11 are suitable constants. From

n similar result holds for X (2) (x). if we suppose that

u

L (  )

1*  max Jac b(t)

Where each

n It follows that X (1) (x)   for each x   . a

t

1

(y) dy.(18)

(3.42)

1* t  C2 ,(19)

n  0,1,..., N  1.



0

n (1)

 1  C1C2  0 For almost every x   , provided that

n X(1) (x)  x  x  x* , For



n (1)

 1  C11*t

Then, we have

n 1

2

n

n X(1) ()

t [0,T]

div b (t, x)  a 0 (x)  0

2

n (1)

 1  tC1 max Jac b  t 

g(x1 )  g(x 2 )  sup . x1  x 2 x l , x 2 

n  u n o X(1) 

n

n X(1) ()

t x  x *

*

We assume that max b(t) t 1.(15) Lip(  )

n 1

2

n det  JacX(1)  (x)  1  t Jac b  t n1 

x1  x 2

u

 dx    u  X (x)  dx   u (y)  det  Jac X  o Y

On the other hand, from (13), we have

 b  t n 1  Lip() x  x* t, Where

2

n u n o X(1)

 b  t n 1 , x   b  t n 1 , x

x,x* x  x*

n

o X(1)



n (1)

n X (1) (x)  x  sup

n



0

 t u n 1

2

u n 1  2 t 2 u 0

 1  C31*t 

n 1 2

n  u n o X (1)



 1  C31*t  2 u n

1 n 1 2 2 0

 t f  t n 1 

0

0 1

n 1

u 0 0  t  1  C31*t  k 1

0

 t f  t n 1  0

n 1 k 2

f tk 

0

(17)

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282

0



 u 0 0  t n 1 max f (t) t [0,T]

0



C  exp  3 1*t n 1  .(21)  2 

Therefore L () -stability holds independently of  . 2

n

The convergence of u to

u  t n  is proven

III.

CONCLUSION

This paper has presented semi- discrete approximation of the parabolic problem. And the stability and convergency of non- stationary advection- diffusion equation is solved by using characteristic Galerkin method.

in a similar way. Defining the error function

òn  u  t n   u n , from (11), we obtain for n  0,1,..., N  1

[1] Adams, R.A., "Sobolev Spaces", Academic Press,

n n u  t n 1   òn 1   u  t n  o X (1)  òn o X (1) 

  u  t n 1   ò

New York, 1975.

t

n 1

[2] Evans, L.C., "Partial Differential Equations",



  div b  t n 1   a 0   u(t n 1 )  òn 1   f  t n 1  ,

 u  t n 1 

t

n 1

 ò

Differential Equations I": Elliptic and Parabolic Equations, (Lecture Notes), 2001.

  div b  t n 1   a 0  ò . n 1

u  t n 1   u  t n 1    div b  t n 1   a 0  u  t n 1   f  t n 1  t

and



u u  t n 1      t n 1   b  t n 1  u  t n 1   . t  t 

Then, we obtain n òn 1  òn o X(1)

t 

 òn 1  div b  t n 1   a 0  òn 1

n u  t n 1   u  t n  o X(1)

in  . Since

[5] Mathews,

J.H., "Numerical Methods for Mathematics", Science and Engineering, PrenticeHall International Education, 1992..

[6] Smith, G.D., "Numerical Solution of Partial

Since 

[3] Griffits, D.F., "An Introduction to Matlab Version [4] Jungel, A., "Numerical Methods for Partial

t   div b  t n 1   a 0  u  t n 1   f  t n 1  

American Mathematical Society, Providence, New York, 1998. 2.2", University of Dundee, Ireland, 1996.

n u  t n 1   u  t n  o X (1)

n òn 1  òn o X(1)

REFERENCES

t



Differential Equations, Finite Difference Methods", 3rd Ed., Clarendon Press, Oxford, 1985.

[7] Süli, E., "Finite Element Methods for Partial Differential Equations", 2002. [8] Thomas, J.W., "Numerical Partial Differential Equations, Conservation Laws and Elliptic Equations", Springer-Verlag, New York, 1998.

u  t n 1   b  t n 1  u  t n 1  (22) t

n X(1) (x)  X  t n ; t n 1 , x   O  (t) 2  ,

it shows that the right hand side in (22) is O ( t). Therefore convergence follows from (21) applied to

òn , recalling that ò0  0.

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