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MAE 555 Non-equilibrium Gas Dynamics. Guest lecturer Harvey S. H. Lam November 16, 2010 On C omputational S ingular P erturbation. Computational Singular Perturbation. Perturbation analysis takes advantage of a small parameter to obtain simplified models and insights .
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MAE 555Non-equilibrium Gas Dynamics Guest lecturer Harvey S. H. Lam November 16, 2010 On Computational Singular Perturbation
Computational Singular Perturbation • Perturbation analysis takes advantage of a small parameter to obtain simplified models and insights. • When the usual perturbation analysis fails, the perturbation analysis which succeeds is called singular perturbation. • Paper and pencil singular perturbation analysis is a collection of tricks. Experience and good judgments about the subject matters are crucial. • Computational singular perturbation is a programmable procedure to general simplified models and insights for a (large) set of first order (non-linear) ordinary differential equations. • No experience or good judgments about the subject matters are needed.
The idea of iteration • How to find x given • How about • How to find y(t) given: • How about
Chemical Kinetics • The equations: where y is a vector of N dimensions, and there are R chemical reactions. Both N and R may be large integers. Usually R>N.
The Objective • To obtain a simplified model with K ODEs and N-K algebraic equations: Want K as small as possible!
Paper and pencil example • N=R=2.
The CSP Challenge • Can we do the same thing for large N and very complicated F(y) without knowing what the small parameter is? • How does one extract physical insights from a mess of computer generated numbers? • For example, what insights are provided when some of the differential equations can be replaced by algebraic equations…
The questions… • What species are unimportant and can be ignored? • What reactions are unimportant and can be ignored? • Which (important) reaction rate constants must be known accurately? • Which (important) reaction rate constant need not be known accurately? • Who is doing what to whom and when?
“Old methods…” • Conservation of atomic elements(also something else may be nearly conserved…) • The quasi-steady approximation on “radicals”.(need to guess who the radicals are…) • The partial equilibrium approximation on fast reactions. (need to guess which reactions are fast)
Initial choice of basis vectors • When you begin, you have all the stoichiometric vectors of all the reactions. You may use N linearly independent stoichiometric vectors for you an’s. • A better idea: use the N right eigenvectors of the NXN J matrix at t=0. Order the eigenmodes in decreasing eigenvalue magnitudes.
If the problem were linear… • The eigenvectors that diagonalize the Matrix will continue to diagonalize the matrix. But we are interested in non-linear problems. So after t=0, the y matrix with constant basis vectors will have off-diagonal terms.
The CSP idea • CSP provides a refinement algorithm to improve the quality of any set of basis vectors. • The quality of a set of basis vectors is the avoidance of fast-slow mode mixing in the solution. A good set has a block diagonalJ matrix. CSP refinement makes the off-diagonal blocks smaller.
CSP refinement algorithm: • Subscript/superscript means refined:
Some closing comments • On validation of models • On the effects of diffusion • On what predictions are unreliable • On usefulness of “slow manifolds” • Quasi-steady vs partial equilibrim… • ILDM • Applications to control theory…
For limited time, these notes are available for downloading at:http://www.princeton.edu/~lam/documents/CSPmae555.pptThese notes do not place emphasis on the mathematical details of CSP which can be found in the published papers.
