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.<br><br>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).
<br><br>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. <br><br>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).<br>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/17096600 |
| Date | 03 December 2021 |
| Creators | Xiaodong Huang (11768345) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Structure_preserving_and_fast_spectral_methods_for_kinetic_equations/17096600 |
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