Singular Perturbation with
Variable Fast Time Scales
Harvey Lam
http://www.princeton.edu/~lam
Princeton University
September, 2007
Why do reduced models?
To reduce the number of unknowns,
 To reduce stiffness,
 To gain insights on the system under
investigation.
-------Finding a “slow manifold” is not enough.

Analytical asymptotics needs a
dimensionless epsilon: <<1

Generic problem statement (t=O(1)):
dy
 g(y;)  w(y), y(0)  y o .
dt

 
What happens when g(y;) is singular in
the small  limit?
(note: w(y) and yo has no singular  dependence)
When g(y;) is uniformly singular in the
small  limit, it is a classical singular
perturbation problem. No problem.
Real world problems often have
no uniformly small epsilons
Real world problems are usually nonlinear
and dimensional.
 Most parameters in real world problems are
dimensional.
 Many interesting real world problems are
intractable by pen-and-pencil analysis.
 All the Rome ODE benchmark problems
have non-uniformly small epsilons.

What do most people do?
(for a N variables problem)
Somehow figure out that M of the original N
variables are fast. Denote them by r.
 The rest of the variables are denoted by s.
 Numerically compute for:

r  S(s)
and call this algebraic relation the Slow
Manifold (useful for certain initial conditions).
Some details (r is fast, s is slow)

Suppose we arrange the variables so that:
r
y   
s

Then the original ODEs are:
dr g r (r, s;) wr (r, s)
  
 

dt s g s (r, s;) ws (r, s)
How useful is any numerical
Slow Manifold r=S(s)?

The original ODEs are:
d r g r (r, s;) wr (r, s)
  
 

dt s g s (r, s;) ws (r, s)
Can we do the following?

r  S(s),
ds
 g s (S(s),s;)  w s (S(s),s).
dt
Answer: sometimes yes,
sometimes no.
Even when epsilon is uniformly small.
 It is yes when the chosen r fast variables
accept the QSSA---quasi-steady-stateapproximation.
 It is no when the r fast variables needs the
PE---partial equilibrium approximation.
 For messy large real world problems, we
usually don’t know which is which.

What is “all you need”?

The leading-order Slow Manifold Projector:
Q slow (M)  I  Q fast (M),
M
Q
fast
(M)   a m bm ,
m1
where am and bm are (column and row)
CSP-refined fast basis vectors.

They are independent of w(y).
 If the CSP bm refinement “converges”, there
is a slow manifold right here.
What the CSP-refined basis
vectors tell you….

Here are the projectors:
Q slow (M)  I  Q fast (M),
M
Q
fast
(M)   a m bm ,
m1
The Slow Manifold after K cycles of 2-step
CSP
refinement is:

f (y;)  b
m
m
g(y;)  O( ), m 1,..., M.
K
The Reduced Model

After the fast transients die (using
Kth-CSP-refined basis vectors):
dy
 Qslow(M) g(y;)  w(y)  O( K ).
dt
One may remove any M differential equations
here and replace them by the M algebraic
 equations of state in the previous slide.
Number of variables is reduced!
The two-step CSP refinement
Step one refines the bm vectors. This
provides the slow manifold.
 Step two refines the am vectors. This
removes stiffness from reduced model.
 If the refinement “iterations” for bm does
not seem to converge, there is no slow
manifold here.

The Williams Problem

dx
 x  xy,
dt
dy 1
 x  ( 1)xy  y
dt 
CSP form of the system (no approximation)


dY(x) 


1

x
x

d x
(1)
dx f (2),
1



f

 
 dY(x) 
dt y 

 




dx
x
Y(x) 
,
  ( 1)x
f (2)
dY (x)
f (x, y;)  Y (x)  y 
,
  ( 1)x dx
(1)
f (2) (x)  x(1 Y (x)),
 (x;) 

.
  ( 1)x
x is non-uniformly small
when  is small.

dY(x) 
d x x  (1) 1 x dx  (2)
  1 f   dY(x) f ,
dt y 
 





dx
x
Y(x) 
,
  ( 1)x
(y is QSSA)
f (2)
dY (x)
f (x, y;)  Y (x)  y 
,
  ( 1)x dx
(1)
f (2) (x)  x(1 Y (x)),
 (x;) 

.
  ( 1)x
The Lindemann Problem
d y 1 z(z  y) 1
  y
   
z
1
dt     
0 
CSP form of the problem (exact):


y   1
(z  y)   




y
1
d

2   2 
  y,
   
1
dt z  1

 
 2 

 (z;)  .
z
Lindemann is a PE problem!

Leading approximation to slow manifold:
y  z  O( )
is completely correct…
but completely useless in the original
ODEs---even if  is uniformly small.

The Semenov ODE Problem
Originals:
d x  x  1 
f (y )


D
(1
x)e
,
  
   a
dt y (1  )y B
y
f (y) 
.
1 y / 
Introduce a new variable:

R  Da (1 x)e f (y ) .
 
x   1 
1
 x 
d  
 B (R  R )  BR  (1  )y,
y

   1 
 


dt




R
  
0

 0

  
More on Semenov ODEs
x   
1 
 x 
d    R  R  


y

B


B
R

(1

)y
,









dt








R 1
0 

 0

Where  plays the role of epsilon:

x(1 y /  ) 2  (1  )(1 x)y
R (x, y) 
,
2
(1 y /  )  B(1 x)
(1 y /  ) 2
 (x, y) 
.
f (y )
2
Da e (1 y /  )  B(1 x)
Coming out of a slow manifold



It is possible for (x,y) to be small for a while,
then become a non-small number later.
Solutions with diverse initial conditions would
become a tight bunch when they enter into the
slow manifold.
When these bunched solutions come out of the
slow manifold, they may still look bunched. But
appearance of bunching is not sufficient to
conclude that there is a slow manifold.
The Semenov PDE Problem

Original PDEs:
dc
 2c
 Le 2   2e /T c,
dt
x
dT  2T
 2   2e /T c.
dt x
CSP form:

d c  1c
   
dt T  
When diffusion is absent
(or if Le=1), T+c
is independent of details
of the chemistry term.
  2c 
Le 2 
  2 x ,  (T)  e( /T 2 ln  ) .
  T 
 x 2 
On the non-chemistry term
How does the magnitude of the diffusion
term depend on the magnitude of ?
 Answer: if diffusion needs to compete (such
as near a boundary), it can and will match
whatever the chemistry term has to offer!
 Whenever this happens, there is no slow
manifold (number of unknowns cannot be reduced).

The Davis Skodje Problem

Original ODEs:
dy1
 y1,
dt
dy 2
( 1)y1  y12
 y 2 
.
2
dt
(1 y1 )
This ODE follows (without approximation):


d 
y1 
y1 
y 2 
  y 2 
,
dt 
1 y1 
1 y1 

which is valid even for arbitrary (y1,y2)!
Large and small 

d 
y1 
y1 
y 2 
  y 2 
,
dt 
1 y1 
1 y1 

Slow manifold: (large  limit)
y1
y2 
, (  ).
1 y1

Conservation Law: (small  limit)

y1
y1 (0)
y2 
 y 2 (0) 
, (  0).
1 y1
1 y1 (0)
Slow manifold versus
Conservation Laws

Consider:
dy
 a1 f 1  ...a M f M  a M 1 f M 1  ... a N f N .
dt

We get a slow manifold when f1 decays
to become small:
1
f (y;)  O().
If fN is always small, AND if bN is the
gradient of a scalar, then we get an
 Conservation Law for
Approximate
that scalar (e.g. an Hamiltonian and …).
Concluding Remarks




In general, finding a slow manifold is not enough.
A slow manifold should be useful for problems
other than the one you found it from.
(i.e. good for different w(y)’s---such as “missing” reactions,
any slow perturbations, diffusions, control forces …)
Reduced models should be able to use single
precision numerical slow manifolds.
Reduced models should tell you some
interesting and insightful things about the system.
http:www.princeton.edu/~lam