Numerical solutions of nonlinear parabolic problems using combined-block iterative methods

This paper is concerned with the block monotone iterative schemes of numerical solutions of nonlinear parabolic systems with initial and boundary condition in two dimensional space. By using the finite difference method, the system is discretized into algebraic systems of equations, which can be represented as block matrices. Two iterative schemes, called the block Jacobi scheme and the block Gauss-Seidel scheme, are introduced to solve the system block by block. The Thomas algorithm is used to solve tridiagonal matrices system efficiently. For each scheme, two convergent sequences starting from the initial upper and lower solutions are constructed. Under a sufficient condition the monotonicity of the sequences, the existence and the uniqueness of solution are proven. To demonstrate how these method work, the numerical results of several examples with different types of nonlinear functions and different types of boundary conditions are also presented

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

Spatial and Physical Splittings of Semilinear Parabolic Problems

Splitting methods are widely used temporal approximation schemes for parabolic partial differential equations (PDEs). These schemes may be very efficient when a problem can be naturally decomposed into multiple parts. In this thesis, splitting methods are analysed when applied to spatial splittings (partitions of the computational domain) and physical splittings (separations of physical processes) of semilinear parabolic problems. The thesis is organized into three major themes: optimal convergence order analysis, spatial splittings and a physical splitting application.In view of the first theme, temporal semi-discretizations based on splitting methods are considered. An analysis is performed which yields convergence without order under weak regularity assumptions on the solution, and convergence orders ranging up to classical for progressively more regular solutions. The analysis is performed in the framework of maximal dissipative operators, which includes a large number of parabolic problems. The temporal results are also combined with convergence studies of spatial discretizations to prove simultaneous space–time convergence orders for full discretizations.For the second theme, two spatial splitting formulations are considered. For dimension splittings each part of the formulation represents the evolution in one spatial dimension only. Thereby, multidimensional problems can be reduced to families of one-dimensional problems. For domain decomposition splittings each part represents a problem on only a smaller subdomain of the full domain of the PDE. The results of the first theme are applied to prove optimal convergence orders for splitting schemes used in conjunction with these two splitting formulations. The last theme concerns the evaluation of a physical splitting procedure in an interdisciplinary application. A model for axonal growth out of nerve cells is considered. This model features several challenges to be addressed by a successful numerical method. It consists of a linear PDE coupled to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. The biological model parameters imply a wide range of scales, both in time and space. Based on a physical splitting, a tailored scheme for this model is constructed. Its robustness and efficiency are then verified by numerical experiments

Numerical solutions of nonlinear elliptic problem using combined-block iterative methods

This thesis is concerned with iterative and monotone methods for numerical solutions of nonlinear elliptic boundary value problems. The methods we study here are called block iterative methods, which solve the nonlinear elliptic problems in twodimensional domain in R2 or higher dimensional domain in Rn. In these methods the nonlinear boundary value problem is discretized by the finite difference method. Two iteration processes, block Jacobi and block Gauss-Seidel monotone iterations, are investigated for computation of solutions of finite difference system using either an upper solution or a lower solution as the initial iteration. The numerical examples are presented for both linear and nonlinear problems, and for both block and pointwise methods. The numerical results are compared and discussed

Adaptive iterative linearization Galerkin methods for nonlinear problems

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed

Numerical Methods for Systems of Nonlinear Parabolic Equations with Time Delays

AbstractThe purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays. The system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations. The mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations. Three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system. A theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained. It is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero