Structure preserving and fast spectral methods for kinetic equations

This dissertation consists of three research projects of kinetic models: a structure preserving scheme for Poisson-Nernst-Planck equations and two efficient spectral methods for multi-dimensional Boltzmann equation.

The Poisson-Nernst-Planck (PNP) equations is widely used to describe the dynamics of ion transport in ion channels. We introduce a structure-preserving semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserving the steady-state).

Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. In this dissertation, we studied the state of the art in the fast Fourier method and discussed its limitation. Next, we proposed a new approach to implement the Fourier method, which can resolve those issues.

However, the Fourier method requires a domain truncation which is unphysical since the collision operator is defined in whole space R^d . In the last part of this dissertation, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we can construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT).