Singular Perturbation with Variable Fast Time Scales Harvey 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)):  What happens when g(y;  ) is singular in the small  limit? ( note: w(y) and y o 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: 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:  Then the original ODEs are:

How useful is any numerical Slow Manifold r=S(s)?  The original ODEs are: Can we do the following?

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-state- approximation.  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: where a m and b m are (column and row) CSP-refined fast basis vectors. They are independent of w(y).  If the CSP b m refinement “converges”, there is a slow manifold right here.

What the CSP-refined basis vectors tell you….  Here are the projectors: The Slow Manifold after K cycles of 2-step CSP refinement is:

The Reduced Model  After the fast transients die (using Kth-CSP-refined basis vectors): 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 b m vectors. This provides the slow manifold.  Step two refines the a m vectors. This removes stiffness from reduced model.  If the refinement “iterations” for b m does not seem to converge, there is no slow manifold here.

The Williams Problem  CSP form of the system (no approximation)

 x  is non-uniformly small when  is small. (y is QSSA)

The Lindemann Problem CSP form of the problem (exact):

Lindemann is a PE problem!  Leading approximation to slow manifold: is completely correct… but completely useless in the original ODEs---even if  is uniformly small.

The Semenov ODE Problem Originals: Introduce a new variable:

More on Semenov ODEs Where  plays the role of epsilon:

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: CSP form: When diffusion is absent (or if L e =1), T+  c is independent of details of the chemistry term.

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: This ODE follows (without approximation): which is valid even for arbitrary  (y 1,y 2 )!

Large and small  Slow manifold: (large  limit) Conservation Law: (small  limit)

Slow manifold versus Conservation Laws  Consider: We get a slow manifold when f 1 decays to become small: If f N is always small, AND if b N is the gradient of a scalar, then we get an Approximate Conservation Law for 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.