Abstract

In this thesis, we present an implementation of a novel collocation method for solving nonlinear parabolic partial differential equations (PDEs) based on triangle meshes. 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 piecewise polynomials in the elements. The polynomial in each element is requiredtosatisfythePDEatcollocationpointsoftheelementandkeepacertaindegreeofcontinuity 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 backsubstituted to solve the variables at the matching points of the sub-regions. This back-substitution is repeated until every element is reached. The accuracy 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. Instead, it may cause singularity. We have given several layouts of specific numbers of collocation and matching points which bring high accuracy. Our solution visualization algorithm directly renders mathematical surfaces instead of any approximation of them. Thus each pixel of the rendered surfaces exactly reflects the corresponding fragment on the mathematical surfaces.