?url_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.title=Constructive+Methods+of+Invariant+Manifolds+for+Kinetic+Problems&rft.creator=Gorban%2C+Prof+Alexander+N.&rft.creator=Karlin%2C+Dr.+Iliya+V.&rft.creator=Zinovyev%2C+Dr.+Andrei+Yu.&rft.subject=Dynamical+Systems&rft.description=We+present+the+Constructive+Methods+of+Invariant+Manifolds+for+model+reduction+in+physical+and+chemical+kinetics%2C+developed+during+last+two+decades.+The+physical+problem+of+reduced+description+is+studied+in+a+most+general+form+as+a+problem+of+constructing+the+slow+invariant+manifold.+The+invariance+conditions+are+formulated+as+the+differential+equation+for+a+manifold+immersed+in+the+phase+space+(the+invariance+equation).+The+equation+of+motion+for+immersed+manifolds+is+obtained+(the+film+extension+of+the+dynamics).++Invariant+manifolds+are+fixed+points+for+this+equation%2C+and+slow+invariant+manifolds+are+Lyapunov+stable+fixed+points%2C+thus+slowness+is+presented+as+stability.+A+collection+of+methods+for+construction+of+slow+invariant+manifolds+is+presented%2C+in+particular%2C+the+Newton+method+subject+to+incomplete+linearization+is+the+analogue+of+KAM+methods+for+dissipative+systems.%0AThe+systematic+use+of+thermodynamics+structures+and+of+the+quasi--chemical+representation+allow+to+construct+approximations+which+are+in+concordance+with+physical+restrictions.%0AWe+systematically+consider++a+discrete+analogue+of+the+slow+(stable)+positively+invariant+manifolds+for+dissipative+systems%2C+invariant+grids.+Dynamic+and+static+postprocessing+procedures+give+us+the+opportunity+to+estimate+the+accuracy+of+obtained+approximations%2C+and+to+improve+this+accuracy+significantly.%0AThe+following+examples+of+applications+are+presented%3A+Nonperturbative+deviation+of+physically+consistent+hydrodynamics+from+the+Boltzmann+equation+and+from+the+reversible+dynamics%2C+for+Knudsen+numbers+Kn~1%3B+construction+of+the+moment+equations+for+nonequilibrium+media+and+their+dynamical+correction+(instead+of+extension+of+list+of+variables)+to+gain+more+accuracy+in+description+of+highly+nonequilibrium+flows%3B+determination+of+molecules+dimension+(as+diameters+of+equivalent+hard+spheres)+from+experimental+viscosity+data%3B+invariant+grids+for+a+two-dimensional+catalytic+reaction+and+a+four-dimensional+oxidation+reaction+(six+species%2C+two+balances)%3B+universal+continuous+media+description+of+dilute+polymeric+solution%3B+the+limits+of+macroscopic+description+for+polymer+molecules%2C+etc.%0A&rft.date=2003&rft.type=Book+Chapter&rft.type=NonPeerReviewed&rft.format=application%2Fpdf&rft.identifier=http%3A%2F%2Fcogprints.org%2F3087%2F1%2FCMIMpro.pdf&rft.identifier=++Gorban%2C+Prof+Alexander+N.+and+Karlin%2C+Dr.+Iliya+V.+and+Zinovyev%2C+Dr.+Andrei+Yu.++(2003)+Constructive+Methods+of+Invariant+Manifolds+for+Kinetic+Problems.++%5BBook+Chapter%5D+++++&rft.relation=http%3A%2F%2Fcogprints.org%2F3087%2F