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

© 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.

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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 .



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    (3)   

220

© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 Here Vh  H10    is a suitable finite-dimensional space and

u 0,h  Vh approximates the initial datum

2u  L2  0, T; L2     . Then t 2

u nh defined in (4) satisfies

for each n  0,1,..., N

u0. k u nh  u  t n    I  1,h  u  t n   exp  C t n 

2.1) A Semi -Implicit Scheme

0

We consider the semi-implicit time-discretization approach. This consists in evaluating the principal part of operator L at time level

t n 1 , whereas the

remaining parts are considered at the time tn. This scheme reads for

0

2  tn 2 u  k k   u 0,h  1,h  u 0   C   I  1,h   s  ds 0 0 t 0  

C  t 

2



tn 0

2  u 2u  s  2 s     t t 1 

n  0,1,..., N  1.

each

1

  2  ds    0  2

(5)

where 1,h is the elliptic projection operator and C  0 is   1  a non-decreasing function of  .    a 0 u hn , v h    f  t n 1  , v h  , v h  Vh ,  (4)  u 0h  u 0,h, Proof   k

1 n 1 n  u h  u h , vh    u nh 1, vh     bi u hn , Di vh  t i 1 n





k Set nh  u nh  1,h u  t n  in (4).

We study semi-discrete methods based on the socalled -scheme:

Then, we obtain

1 n 1 n  u h  u h , vh   a  u nh 1  1   u nh , vh    f  t n 1   1    f  t n  , vh  , vh Vh ,  t   u 0h  u 0,h , 





1 n 1 k h  1,h  u  t n 1    hn  1,hk  u  t n   , vh t

for each n  0, 1, ..., N  1, where 0    1, t  T is the N

time step, N is a positive integer, and u 0,h  Vh is a suitable approximation of the initial datum u0.

1.     2  k The elliptic projection operator 1,h is defined as



k 1,h

vV

 v   Vh

implies a  

k 1,h

 v  , vh   a  v, vh 

k   bi hn  bi 1,h  u(t n )  , Di v h n

i 1

This includes the schemes: forward Euler    0  , backward Euler    1 and Crank-Nicolson

follows: For each







k  a 0hn  a 0 1,h  u  t n  , vh  f  t n1  , vh .

That is, n 1 n 1 h  nh , vh     hn 1 , vh     bi hn , Di vh    a 0hn , vh   t i 1 1 k   f  t n 1  , vh   1,h  u  t n 1   u  t n   , vh t





k   1,h  u  t n 1   , vh

vh  Vh .

 

k  nh 1   1,h  u  t n 1   , vh

n







 



k  bi 1,h  u  t n  , Di vh  a 0 1,hk  u  t n  , vh , i 1

2.2 Theorem Assume that u 0  H10    and that the solution u to (1) is

such

that

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u  L2  0, T; H10     and t

1 n 1  h  nh , vh    nh 1, vh  t   bi hn , Di vh    a 0hn , vh  n

i 1

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© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880  u  1 k    t n 1  , v h   1,h  u  t n 1   u  t n   , v h  t  t

 f  t  , v   1t    u  t   u  t   , v    u  t  , v  

n 1

k 1,h

h

n 1

n



h



   bi u  t n  , Di vh    a 0 u  t n  , vh     u  t n  , vh    a   u  t   , v        u  t   , v  i 1      u  t   , v       u  t   , v     b u  t  , D v    b   u  t   , D v    a   u  t   , v  , n 1



n

k   bi 1,h  u  t n   , Di v h

h

i 1

n

k 1,h

0

k 1,h

n

k 1,h

n

h

i 1

i

n

n 1

i

h

k 1,h

h

n

n

h

k 1,h

n 1

h

n

k 1,h

i

i 1

n 1

i

h

0

h

1 n 1  h  hn , vh    hn1, vh  t

(6)

  bi hn , Di v h    a 0nh , v h     nh , v h  , (7) n

On the other hand, the solution u satisfies

i 1

where  h  Vh is defined by the relation

 u    t n 1  , v h    f  t n 1  v h     u  t n 1  , v h   t 

n

u t u t    , V    ut  t    t   , v

n

   bi u  t n 1  , Di v h    a 0 u  t n 1  , v h  .

n 1

n h

n

n 1

h

 1  t n 1 k     I  1,h  ut  s  ds, vh  t  t n

i 1

Thus, ( 6 ) becomes

h

  

n 1 n 1 h  nh , vh     hn 1 , vh     bi hn , Di vh    a 0hn , vh   t i 1

n (8)  u  1 k    t n 1  , vh   1,h u  t n 1   u  t n   , vh    bi u  t n  , Di vh    t  t   i 1





Therefore h satisfies a scheme like ( 4 ). we have n



k   a 0 u  t n  , vh     u  t n  , vh    1,h  u  t n  , vh

n



 



k  bi 1,h  u  t n 1   , Di vh  a 0 1,hk  u  t n 1   , vh . i 1

Let us further recall that the definition of

 reads k 1,h

n

   u  t n ), vh      bi u  t n  , Di vh    a 0 u  t n  , vh  i 1



 k   u 0,h  1,h  u0  0  2  t  exp C1 n 1  a 0 L      nh

2



2 0



Cl m 1 n t   h  n 0

2 1,h

  (9) 

 .

The first term at the right hand side of (8) can be estimated as follows: u  t n 1   u  t n   u  1  t n1  2u , v h      s  t n  2  s ds, v h    t n 1   tn  t  t  t  t      2u  1 t   s  t n   2  s  , v h  ds t  t  t  n 1





n



 



k     1,h  u  t n  , vh   bi 1,hk  u  t n  , Di vh  a 0 1,hk  u  t n  , vh .

Since

i 1

 k commutes with 1,h ( 6) can be rewritten t



n

1 t n1 2u s  t n  2 s    t t n t

v h 1 ds. 1,h

as n The second term at the right hand side of (8 ) can be 1 n 1 h  hn , vh     hn 1 , vh     binh , Di v h    a 0nh , v h   estimated as follows: t i 1

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© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 1  t n1 u u  1 t n1 k k   t n  I  1,h   s  ds, vh    t n  I  1,h   s  t  t t  t

1,h

vh 1 ds.

  b   u  t   u  t  , D v   n

k 1,h

i

i 1

n 1

n

i

k 1,h  u  t n   u  t n 1 

h

k  1,h  u  t n   u  t n 1  

vh

0

1,h

vh

2

1  t n1 u k    t  I  1,h   s  t t  n

2



Moreover, we have

1

 t  t n1  2 u   s 2   t  3  n t

n 2 h 1,h

2 ds   1,h  1

2 ds   1,h 

1

2

 

k 1,h

 u  t   u  t  n 1

n

1

0

 a0

L (  )



k 1,h

 u  t   u  t  n 1

n

  ,  

0

and

a

0

 u  t   u  t  , v    u  t   u  t  v k 1,h  u  t n   u  t n 1   0 vh L ()

k 1,h

 a0

 a0

n 1

n

k 1,h

L (  )



h

n 1

n

h 1

1,h



1

.



h

u s  t 2

n

tn

1 t n1 u k I  1,h s     t t n t

k  1,h  u  t n 1   u  t n  

 , v   n h

h

vh

1

0

1,h

vh 1  a 0

0

vh 1 ds

k 1,h  u  t n 1   u  t n  

dx  1,h

vh 1 ,

0

1 t n1  I  1,hk  ut s  t  t n

L (  )



1 a0

ds

nh

1,h

k 1,h  u  t n 1   u  t n   .

2 1,h

0

2 L (  )



t 

t n 1 tn



2 L   



t 

t n 1 tn

By using inequality, we have

 sup

v h Vh vh  0

 , v   vh

h

1

1 t



t n 1 tn

s  t n 

2

ds



1 2

 t n1  2 u  s  t n t 2 



1 t

  ds   t n 1

tn

t n 1 tn

 I    ut  s 

k  1,h  u  t n 1   u  t n  

 a0

L (  )

2 L   

k 1,h

0

k 1,h  u  t n 1   u  t n   ,

1

 ds   1,h  2

1 2

k 1,h

2 0

 ,  

2  u (s) ds  , t  0

since the operator

2  u s   ds  , t  1

 k is uniformly bounded in

2 0

2 m 1  2 CC 1 t n1 u k k   u 0,h  1,h  u 0  0  1 2 t    tn  I  1,h  s  ds   t n  0  t  0

 C3 t 

t n 1 tn

2  u 2u  s   2 s    t t 1 

     t ds   exp C1 n 1  a 0      0 



2

2 L   

2 2 2 k u nh  u  t n    I  1,h  u  t n  0  exp  C t n   u0,h  1,hk  u0  0 0 

0

 C 

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k 1,h  u  t n 1   u  t n 

ds 1,h

1,h 2 ds  H1  . From (7), we obtain  0  1,h  2

nh 1 2

2

0

2

2 2 2  t n 1  u 1 t n1 u k  C2  t  s ds  I   s  1,h  t   ds 2   t  t n  t n t 0 0

1 a0

1,h

1,h

1 t n1  I  1,hk  ut s  t  t n

1,h

vh 1 ds

L (  )

k  1,h  u  t n 1   u  t n    a 0

 nh

ds 

+

1 t n1 2u s  t n  2 s   t t n t

n h

2

2 2 2  t n 1  u 1 t n1 u k  C2  t  s ds  I   s ds       1,h 2 t  t n t  t n t 0 0

2

t n 1

n h

2  t n 1  u  C2  t  s 2    t n t

k  1,h  u  t n 1   u  t n   a 0

Therefore, (8) becomes

  , v   1t  s  t 

n 2 h 1,h

tn 0

 I    ut  s  ds 0 2

k 1,h

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223

 ,

© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 2  u 2u  C  t    s   2 s   0  t t 1  k where nh  u nh  1,h  u  t n .

2

tn

2

0

  ds  

 , 

k u nh  u  t n    I  1,h  u  t n   exp  C t n  0

 C 

tn 0

 C  t 

0

 I  1,hk  2



tn 0

u

0,h

k  1,h u0 

2 0

u  s  ds t 0 2

2  u 2u  s   2 s    t t 1 

III.

1

  2  ds  .   0  2

CONCLUSION

This paper has presented parabolic advectiondiffusion problem. Then, we study numerical schemes for initial boundary value problem. And then, we prove convergence of the semi implicit problem.

REFERENCES [1] [2] [3]

[4]

[5]

[6]

[7] [8]

Adams, R.A., "Sobolev Spaces", Academic Press, New York, 1975.

Evans, L.C., "Partial Differential Equations", American Mathematical Society, Providence, New York, 1998.

Griffits, D.F., "An Introduction to Matlab Version 2.2", University of Dundee, Ireland, 1996.

Jungel, A., "Numerical Methods for Partial Differential Equations I": Elliptic and Parabolic Equations, (Lecture Notes), 2001.

Mathews, J.H., "Numerical Methods for Mathematics", Science and Engineering, Prentice-Hall International Education, 1992..

Smith, G.D., "Numerical Solution of Partial Differential Equations, Finite Difference Methods", 3rd Ed., Clarendon Press, Oxford, 1985.

Süli, E., "Finite Element Methods for Partial Differential Equations", 2002.

Thomas, J.W., "Numerical Partial Differential Equations, Conservation Laws and Elliptic Equations", Springer-Verlag, New York, 1998.

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