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```					IMEX Methods
for
Equations

Speaker : Volha Shchetnikava

Eindhoven
2008
Contents
1.   Introduction
    A-D-R equations

2.   Implicit-explicit (IMEX) methods
     Description
     Stability of IMEX methods
     Why IMEX?

3.   IMEX linear multistep methods
      Design of IMEX linear multistep methods
      Examples

4.       Numerical Experiments

5.   Conclusions
1. Introduction

Model problem

ut  (au) x  (du x ) x  f (u )           ( x, t )  Ω  [0, T]
 nu  g N                        ( x, t )  Ω N  [0, T]
u ( x, t )  g D                  ( x, t )  Ω D  [0, T]
u ( x,0)  x0                     x  Ω ,        Ω  0,1

where   u(x,t)   -   concentration of a certain species,
a(x,t)   -   velocity of flowing medium,
d(x,t)   -   diffusion coefficient
f(u)     -   source, sink function
1. Introduction

Fields of application

   Environmental modeling (weather forecast, water flow)
   Mathematical biology (bacterial growth, tumor growth)
   Chemistry
   Mechanics
1. Introduction

Using numerical technique is The Method of Lines (MOL).

MOL algorithm:
1.   Discretize all spatial operators
2.   Obtain a system of ODEs

w(t )  F (t , w(t )) t  0 w(0)  w0

3.   Integrate ODEs system in time

1.   Spatial discretization and time integration are treated separately
2.   Spatial discretization - easy to combine different schemes
3.   Time integration - free to choose suitable method
2. Implicit-Explicit Methods
Description

IMEX method - different integrators to different terms.

System of ODEs

w(t )  F (t , w(t ))  F0 (t , w(t ))  F1 (t , w(t ))

F0 is a non - stiff term
F1 is a stiff term

Often,
F1 emanates from reaction - diffusion terms
2. Implicit-Explicit Methods
Description

IMEX -  Method

wn 1  wn   F0 (t n , wn )  (1   )  F1 (t n , wn )    F1 (t n 1 , wn 1 )

1
where 
2

Explicit Euler  A - stable implicit  - method
2. Implicit-Explicit Methods
Description

Inserting the exact solution w(t ) gives the temporal truncation error

1
 n  (   )w(t n )   (t n )  ( 2 ), where (t )  F0 (t , w(t ))
2

With stationary solution we have a zero truncation error.
2. Implicit-Explicit Methods
Stability of IMEX method

Test equation
w(t )  0 w(t )  1w(t ) and let z j   j , j  0,1

Stability expl. method for 0 
  Stability of the IMEX scheme
Stability impl. method for 1 

Applicatio n of IMEX -  method yields

wn 1  Rwn , R  R( z0 , z1 )

1  z0  (1   ) z1
R( z0 , z1 ) 
1   z1
Stability requires R( z0 , z1 )  1
2. Implicit-explicit (IMEX) methods
Stability of IMEX methods

D0  z0  C : the IMEX schemeis stable for any z1  C  
D1  z1  C : the IMEX schemeis stable for any z0 S 0 
2. Implicit-explicit (IMEX) methods
Stability of IMEX methods

0 and 1 are independent  F0 (au x ) and F1 (du zz ) act in different directions
If 0 and 1 are dependent  Different results are obtained
The implicit t reatment of 1 can stabilize the process

ut  au x  du xx
z0  iv sin(2 ),        z1  4  sin 2 ( )
with v  a / h,   d / h 2

R  1 iff v 2  2  and 2(1   )   1
Condition for stability is
1
     and   2d / a 2
2
2. Implicit-explicit (IMEX) methods
Why IMEX?

Why not fully explicit method?
   Stability will require very small step sizes for stiff sources

Why not fully implicit method?
   For advection descretizations the implicit relations are hard to solve
   High computational cost

Why IMEX?
   IMEX show a significant computational savings due to less restricted time step size
   The method remains stable for time steps much larger than those that would be possible for a purely
explicit method.
   Very effective in many situations
   Easy to apply
3. IMEX linear multistep methods
Design of IMEX linear multistep methods

Fully impicit linear k - step method

k                        k

 w
j 0
j   n j       j ( F0 (t n  j , wn  j )  F1 (t n  j , wn  j ))
j 0

Explicit F0 can be derived by extrapolation formula

k
 (t n  k )    j (t n  j )  ( q ), where  (t )  F0 (t , w(t ))
j 0

This leads to the k - step IMEX method

k                       k 1                           k

 w
j 0
j   n j       F0 (t n  j , wn  j )    j F1 (t n  j , wn  j ), with  *   j   k  j
j 0
*
j
j 0
j
3. IMEX linear multistep methods
Examples

Explicit midpoint rule (Leap - Frog)
wn 1  wn 1  2F (t n , wn )

Trapezoidal rule (Crank - Nicolson)
wn 1  wn 1   ( F (t n 1 , wn 1 )  F (t n 1 , wn 1 ))

IMEX - CNLF scheme
wn 1  wn 1  2F0 (t n , wn )  F1 (t n 1 , wn 1 )  F1 (t n 1 , wn 1 )

.
3. IMEX linear multistep methods
Examples

Implicit two - step BDF
3              1
wn 1  2wn  wn 1  Fn 1
2              2

Explicit two - step BDF
3              1
wn 1  2wn  wn 1  2Fn  Fn 1
2              2

IMEX - BDF scheme
3              1
wn 1  2wn  wn 1  2F0 (t n , wn )  F0 (t n 1 , wn 1 )  F1 (t n 1 , wn 1 )
2              2
3. IMEX linear multistep methods
Examples

3                1
wn 1  wn  F (t n , wn )  F (t n 1 , wn 1 )
2                2

IMEX scheme
3                   1
wn 1  wn  F0 (t n , wn )  F0 (t n 1 , wn 1 )  F1 (t n 1 , wn 1 ) 
2                   2
3                            1
 (  2 )F1 (t n , wn )  (  )F1 (t n 1 , wn 1 )
2                            2

1
The implicit method is A - stable if  
2
1
     - trapezoidal rule
2
4. Numerical experiments

ut  au x  u xx  u (1  u )   ( x, t )  (0,1  [0, T]
)
u x (0, t )  0                  t  [0, T]
u (1, t )  (1  sin(t )) / 2           t  [0, T]
u ( x , 0)  1 / 2               x  0,1
4. Numerical experiments

a  1,   0.01,   10,   10

f0_unst2.avi
f1_unst1.avi

a  5,   10,   10,   10
f0_st1.avi
f1_st2.avi

a  25,   10,   10,   10

f1_25_2.avi
5. Conclusions

   Significant computational saving
   Stable for time steps larger then for explicit method
   IMEX schemes are not universal for all problems
   Very effective in many situations
   IMEX BDF is more stable then IMEX -CNLF
Thank you!

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