Master Thesis Defense: Xu Chen

Speaker: Xu Chen

Supervisor: Dr. E. J. Doedel

Examining Committee:
Drs. T. D. Bui, A. Krzyzak, W. Shang (Chair)

Title: Collocation Methods for Nonlinear Parabolic Partial Differential Equations

Date: Tuesday, April 4, 2017

Time: 13:00

Place: EV 3.309

ABSTRACT

In this thesis, we present an implementation of a novel collocation method for solving nonlinear parabolic partial differential equations (PDEs). The temporal partial derivative is discretized using the implicit Euler-backward finite difference scheme. The spatial domain of the PDEs discussed in this thesis is two-dimensional. The domain is first triangulated and then refined into appropriately sized triangular elements by the Rivara algorithm. The solution is approximated by the piecewise polynomials in the elements. The polynomial in each element is required to satisfy the PDE at collocation points of the element and keep a certain degree of continuity with the polynomials in the neighboring elements via matching points. Nested dissection is used recursively, from the elements up to the entire domain, to merge all pairs of sibling sub-regions for eliminating the variables at the matching points on the common sides shared by the merged sub-regions. Then by applying global boundary conditions, we solve for the solution values at the boundary points of the entire domain. The solutions at the boundary points of the domain are back-substituted to solve the variables at the matching points of the sub-regions. This back-substitution is repeated until every element is reached. The error of the solution is affected by the time step, granularity of the subdivision, the number and location of matching points, and the number and location of collocation points. Increasing the number of matching points or collocation points does not always improve the accuracy. On the other hand, it may cause singularity. Our mesh generation method can be used for complex polygonal domains with adaptive mesh refinement. Optimization of computational performance is also described in this thesis. Our visualization of the solution makes use of state-of-the-art 3D computer graphics technologies, such as WebGL, GPU programming, etc., to render pixel-accurate polynomial surfaces. The visualization program is platform independent.