Monotone iterative methods for solving nonlinear partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand

A key aspect of the simulation process is the formulation of proper mathematical models. The model must be able to emulate the physical phenomena under investigation. Partial differential equations play a major role in the modelling of many processes which arise in physics, chemistry and engineering. Most of these partial differential equations cannot be solved analytically and classical numerical methods are not always applicable. Thus, efficient and stable numerical approaches are needed. A fruitful method for solving the nonlinear difference schemes, which discretize the continuous problems, is the method of upper and lower solutions and its associated monotone iterations. By using upper and lower solutions as two initial iterations, one can construct two monotone sequences which converge monotonically from above and below to a solution of the problem. This monotone property ensures the theorem on existence and uniqueness of a solution. This method can be applied to a wide number of applied problems such as the enzyme-substrate reaction diffusion models, the chemical reactor models, the logistic model, the reactor dynamics of gasses, the Volterra-Lotka competition models in ecology and the Belousov-Zhabotinskii reaction diffusion models. In this thesis, for solving coupled systems of elliptic and parabolic equations with quasi-monotone reaction functions, we construct and investigate block monotone iterative methods incorporated with Jacobi and Gauss--Seidel methods, based on the method of upper and lower solutions. The idea of these methods is the decomposition technique which reduces a computational domain into a series of nonoverlapping one dimensional intervals by slicing the domain into a finite number of thin strips, and then solving a two-point boundary-value problem for each strip by a standard computational method such as the Thomas algorithm. We construct block monotone Jacobi and Gauss-Seidel iterative methods with quasi-monotone reaction functions and investigate their monotone properties. We prove theorems on existence and uniqueness of a solution, based on the monotone properties of iterative sequences. Comparison theorems on the rate of convergence for the block Jacobi and Gauss-Seidel methods are presented. We prove that the numerical solutions converge to the unique solutions of the corresponding continuous problems. We estimate the errors between the numerical and exact solutions of the nonlinear difference schemes, and the errors between the numerical solutions and the exact solutions of the corresponding continuous problems. The methods of construction of initial upper and lower solutions to start the block monotone iterative methods are given

Mathematical analysis of a model of river channel formation.

The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin

Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime

A multidimensional hydrodynamic code for structure evolution in cosmology

A cosmological multidimensional hydrodynamic code is described and tested. This code is based on modern high-resolution shock-capturing techniques. It can make use of a linear or a parabolic cell reconstruction as well as an approximate Riemann solver. The code has been specifically designed for cosmological applications. Two tests including shocks have been considered: the first one is a standard shock tube and the second test involves a spherically symmetric shock. Various additional cosmological tests are also presented. In this way, the performance of the code is proved. The usefulness of the code is discussed; in particular, this powerful tool is expected to be useful in order to study the evolution of the hot gas component located inside nonsymmetric cosmological structures.Comment: 34 pages , LaTex with aasms4.sty, 7 postscript figures, figure 4 available by e-mail, tared , gziped and uuencoded. Accepted Ap

Identification of Chemotaxis Models with Volume Filling

Chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant. A crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant. In this paper, we investigate the identification of these non-linear parameter functions in a chemotaxis model with volume filling. We also discuss the numerical realization of Tikhonov regularization for the stable solution of the inverse problem. Our theoretical findings are supported by numerical tests.Comment: Added bibfile missing in v2, no changes on conten

Convergence of the Crank-Nicolson-Galerkin finite element method for a class of nonlocal parabolic systems with moving boundaries

The aim of this paper is to establish the convergence and error bounds to the fully discrete solution for a class of nonlinear systems of reaction-diffusion nonlocal type with moving boundaries, using a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of any degree. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with some existing moving finite elements methods are investigated