Fourth-Order Compact Finite Difference Methods And Monotone Iterative Algorithms For Boundary Value Problems

Boundary value problems for differential equations arise in various fields of engineering and physical processes. It is both theoretically and practically important to propose effective numerical methods for such problems. The main purpose of this thesis is to propose a fourth-order compact finite difference method for a class of semi-linear2nth-order multi-point boundary value problems and a class of two-dimensional semi-linear elliptic boundary value problems with variable coefficients, respectively, and to develop some effective monotone iterative algorithms for the numerical solutions.This thesis is mainly divided into two parts. In the first part, we first give a numerical treatment for a class of semi-linear second-order multi-point boundary value problems. The problems are discretized by the fourth-order compact Numerov's method. The exis-tence and uniqueness of the numerical solutions are investigated by the method of upper and lower solutions. The convergence and the fourth-order accuracy of the method are proved. An accelerated monotone iterative algorithm with the quadratic rate of conver-gence is developed for solving the resulting nonlinear discrete problems. Some applications and numerical results are given to demonstrate the high efficiency of the approach. Then, we propose a fourth-order compact finite difference method for a class of semi-linear2nth-order multi-point boundary value problems. The existence and uniqueness of the finite difference solutions are investigated by the method of upper and lower solutions. The convergence and the fourth-order accuracy of the method are proved. An efficient lin-ear monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems. Various sufficient conditions for the construction of upper and lower solutions are obtained. All theoretical analyses don't impose any monotone requirement on the nonlinear term. Some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach.In the second part, a fourth-order compact finite difference method with non-isotropic mesh is proposed for a class of two-dimensional semi-linear elliptic boundary value prob-lems with variable coefficients. The domain is a union of rectangular domains that includes various nonstandard regions such as an L-shaped or a T-shaped region. The existence and uniqueness of the finite difference solutions are investigated by the method of upper and lower solutions. The convergence and the fourth-order accuracy of the proposed method are proved with respect to the discrete L∞-norm. Some explicit discrete H1-, H2-and L2-norm error estimates are also derived for the special case of the constant coefficients. A Picard type of monotone iterative algorithm, using either an upper solution or a lower solution as the initial iteration, is presented for solving the resulting nonlinear system effi-ciently. Numerical results verify the theoretical results and justify the use of non-isotropic mesh. The effectiveness of the proposed method on nonstandard domains is illustrated through two problems in L-shaped and T-shaped domains.Then, we develop a unified monotone iterative algorithm for the above discrete com-pact finite difference system. This unified algorithm includes the often used Picard method and the pointwise and block methods of Jacobi and Gauss-Seidel, and also gives a number of new and efficient block monotone iterative methods that are appropriate for parallel computing. The produced iterative sequence converges monotonically and geometrically to a maximal solution or a minimal solution of the compact difference system. Theoretical comparison results among the various monotone iterative methods are obtained. These comparisons demonstrate that the block methods converge faster than the corresponding pointwise methods, and are more preferable for numerical computations. Numerical re-sults by the monotone iterative methods are given for three model problems, including a problem in a T-shaped domain, and are compared with the known analytic solution for accuracy. Also compared are the rates of convergence of the block methods as well as the pointwise methods.Finally, a higher-order monotone iterative algorithm is proposed to solve the above discrete compact finite difference system. It yields two sequences of iterations that con-verge monotonically from above and below, respectively, to a maximal or a minimal solution of the compact difference system. The monotone convergence property of the iterations gives concurrently improving upper and lower bounds for the solution in each iteration. The rate of convergence for the sum of the two sequences is of order p+2, where p is a nonnegative integer depending on the construction of the algorithm. Applications to two model problems, including a problem in a T-shaped domain, give some numerical results that illustrate the effectiveness of the proposed algorithm.