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.

We transform the density function as the Hopf-Cole transformation and expand the phase function in a power series about the Knudsen number.

This is similar to moment closure methods for similar equations but we make no assumptions about the moments of the kinetic equation.

The leading order term of the power series is the viscosity solution of a particular Hamilton-Jacobi equation.

The Hamilton-Jacobi equation involves an implicitly-defined Hamiltonian which is embedded in an integral with respect to the velocity variables.

The Hamiltonian is the root of a nonlinear integral equation so Gaussian quadrature and Newton's method are used to recover it.

The first order term in the power series is related to a transport equation.

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.

The integrals can be evaluated efficiently using Gaussian quadrature.

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.