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

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>