A Hopf-Cole transformation based asymptotic method for kinetic equations with a BGK collision operator in the large scale hyperbolic limit

dc.contributor.advisor Songting Luo
dc.contributor.author Payne, Nicholas
dc.contributor.department Mathematics
dc.date 2018-08-11T08:14:32.000
dc.date.accessioned 2020-06-30T03:06:27Z
dc.date.available 2020-06-30T03:06:27Z
dc.date.copyright Fri Jan 01 00:00:00 UTC 2016
dc.date.embargo 2001-01-01
dc.date.issued 2016-01-01
dc.description.abstract <p>We propose an effecient asymptotic approach to approximating the density function of kinetic equations with Bhatnagar-Gross-Krook (BGK) relaxation operators. These types of equations are relevant in the study of particle dynamics in thermodynamic systems; gas dynamics for example. We consider a simplified BGK operator in this thesis for the purposes of explaining the proposed method which has the potential to deal with more general BGK operators.</p> <p>We transform the density function as the Hopf-Cole transformation and expand the phase function in a power series about the Knudsen number.</p> <p>This is similar to moment closure methods for similar equations but we make no assumptions about the moments of the kinetic equation.</p> <p>The leading order term of the power series is the viscosity solution of a particular Hamilton-Jacobi equation.</p> <p>The Hamilton-Jacobi equation involves an implicitly-defined Hamiltonian which is embedded in an integral with respect to the velocity variables.</p> <p>The Hamiltonian is the root of a nonlinear integral equation so Gaussian quadrature and Newton's method are used to recover it.</p> <p>The first order term in the power series is related to a transport equation.</p> <p>Both the Hamilton-Jacobi equation and the transport equation are formulated in the physical space with necessary components defined as integrals with respect to the velocity variables.</p> <p>The integrals can be evaluated efficiently using Gaussian quadrature.</p> <p>We use well-established techniques for time-dependent Hamilton-Jacobi equations to solve these two equations and recover an estimate of the phase function. We then transform the phase function back to a faithful estimate of the original density function.</p>
dc.format.mimetype application/pdf
dc.identifier archive/lib.dr.iastate.edu/etd/15788/
dc.identifier.articleid 6795
dc.identifier.contextkey 11165317
dc.identifier.doi https://doi.org/10.31274/etd-180810-5416
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath etd/15788
dc.identifier.uri https://dr.lib.iastate.edu/handle/20.500.12876/29971
dc.language.iso en
dc.source.bitstream archive/lib.dr.iastate.edu/etd/15788/Payne_iastate_0097M_15845.pdf|||Fri Jan 14 20:46:34 UTC 2022
dc.subject.disciplines Applied Mathematics
dc.subject.keywords BGK
dc.subject.keywords equation
dc.subject.keywords Hamilton-Jacobi
dc.subject.keywords Hopf-Cole
dc.subject.keywords hyperbolic
dc.subject.keywords kinetic
dc.title A Hopf-Cole transformation based asymptotic method for kinetic equations with a BGK collision operator in the large scale hyperbolic limit
dc.type article
dc.type.genre thesis
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48
thesis.degree.discipline Applied Mathematics
thesis.degree.level thesis
thesis.degree.name Master of Science
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