New analysis of linearized numerical schemes for several nonlinear convection-diffusion equations and Schrödinger type system

Title: New analysis of linearized numerical schemes for several nonlinear convection-diffusion equations and Schrödinger type system Authors: Wang, Jilu (王冀魯) Abstract: The thesis is concerned with a new analysis of linearized numerical schemes for several nonlinear convection-diffusion equations and Schrödinger type system. The method of characteristics is especially effective for convection-dominated diffusion problems. Due to the nature of characteristic temporal discretization, the method allows one to use a large time step in many practical computations, while all previous theoretical analyses always required certain restrictions on the time stepsize. The first part of the thesis is to present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with finite element approximation. Here, we consider two physical models: incompressible miscible flow in porous media and the time-dependent Navier-Stokes equations. For this purpose, we introduce a characteristic time-discrete system. We prove that the L2 error bound of the fully discrete method of characteristics to the time-discrete system is τ-independent and the numerical solution is bounded in W1,∞-norm unconditionally, where τ denotes the time stepsize. With the boundedness, optimal error estimates are established in a traditional manner. Secondly, we study linearized Crank-Nicolson finite element methods (FEMs) and finite difference methods (FDMs) for Schrödinger type system. We obtain optimal L2 error estimates without any time-step restrictions. Our approaches are based on an error splitting technique for FEMs, and a rigorous analysis in both real and imaginary parts of the error functions for FDMs. The third part of this thesis is concerned with mathematical modeling, analysis and computations of heat and sweat transport in fibrous media with a non-local thermal radiation and phase change. The model, based on a combination of these classical heat transfer mechanisms (convection, conduction and radiation), is governed by a nonlinear, degenerate and strongly coupled parabolic system. We prove the global existence of positive/non-negative weak solutions of this nonlinear parabolic system. A typical clothing assembly with a polyester batting material sandwiched in two laminated covers is investigated numerically. Numerical results show that the contribution of radiative heat transfer is comparable with that of conduction/ convection in the sweating system. Notes: CityU Call Number: QA377 .W34 2015; ix, 157 pages : illustrations 30 cm; Thesis (Ph.D.)--City University of Hong Kong, 2015.; Includes bibliographical references (pages 141-157)