IMEX Methods
for
Advection-Diffusion-Reaction
Equations
Speaker : Volha Shchetnikava
Adviser: dr.ir.J.H.M. ten Thije Boonkkamp
Eindhoven
2008
Contents
1.
Introduction

2.
Implicit-explicit (IMEX) methods



3.
A-D-R equations
Description
Stability of IMEX methods
Why IMEX?
IMEX linear multistep methods


Design of IMEX linear multistep methods
Examples
4.
Numerical Experiments
5.
Conclusions
1. Introduction
Advection-Diffusion-Reaction Equations
Model problem
ut  (au ) x  (du x ) x  f (u )
where
( 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 ,
u(x,t)
a(x,t)
d(x,t)
f(u)
-
concentration of a certain species,
velocity of flowing medium,
diffusion coefficient
source, sink function
Ω  0,1
1. Introduction
Advection-Diffusion-Reaction Equations
Fields of application

Environmental modeling (weather forecast, water flow)

Mathematical biology (bacterial growth, tumor growth)

Chemistry

Mechanics
1. Introduction
Advection-Diffusion-Reaction Equations
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
Advantages of MOL:
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,
F0 emanates from advection term
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 )
where  
1
2
Explicit Euler  A - stable implicit  - method
2. Implicit-Explicit Methods
Description
Inserting the exact solution w(t ) gives the temporal truncatio n error
1
2
 n  (   )w(t n )   (t n )  ( 2 ), where  (t )  F0 (t , w(t ))
With stationary solution w e have a zero truncatio n 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 )
R ( z0 , z1 ) 
1  z0  (1   ) z1
1   z1
Stability requires R ( z0 , z1 )  1
2. Implicit-explicit (IMEX) methods
Stability of IMEX methods

D0  z0  C : the IMEX scheme is stable for any z1  C 

D1  z1  C : the IMEX scheme is stable for any z0 S 0 
2. Implicit-explicit (IMEX) methods
Stability of IMEX methods
0 and 1 are independen t  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
 w
j 0
j
n j
k
    j ( F0 (t n  j , wn  j )  F1 (t n  j , wn  j ))
j 0
Explicit F0 can be derived by extrapolat ion 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
 w
j 0
j
n j
k 1
k
    F0 (t n  j , wn  j )    j F1 (t n  j , wn  j ), with  *j   j   k  j
j 0
*
j
j 0
3. IMEX linear multistep methods
Examples
Explicit midpoint rule (Leap - Frog)
wn 1  wn 1  2F (t n , wn )
Trapezoida l 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 t wo - step BDF
3
1
wn 1  2 wn  wn 1  Fn 1
2
2
Explicit t wo - step BDF
3
1
wn 1  2 wn  wn 1  2Fn  Fn 1
2
2
IMEX - BDF scheme
3
1
wn 1  2 wn  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
Explicit Adams method
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
The implicit method is A - stable if  

1
- trapezoid al rule
2
1
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
adams_5_2.avi
a  25,   10,   10,   10
f1_25_2.avi
adams_25_3.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!