New block iterative methods for the numerical solution of boundary value problems
The work presented in this thesis is wholly concerned with the derivation of a new group technique for the solution of boundary value problems using finite difference approximations. The thesis commences with a general description and classification of partial differential equations and its related discretised matrix, also a description of some physical problems which involve elliptic partial differential equations are given. The numerical solution of linear elliptic partial differential 'equations by the method of finite differences always leads to a large number of linear algebraic equations, the determination of the set of linear equations corresponding to an elliptic partial differential equation is shown and different methods for their solution are described. An introduction to the finite element technique is also included as an alternative to the finite difference method of solution. In Chapter 4, we present the solution of the linear system of equations by group iterative methods. New strategies are established concerning the novel approach of using a small group of points of fixed size. The 2,4,6,9,12,16 and 25 point group structure is proposed, developed and analysed theoretically and experimentally. From these results an analysis of the computational complexity of an optimum group structure can be determined and it can be deduced that such splittings can be both useful and efficient. In a similar manner, an explicit 8 point group is used to solve the elliptic partial differential equations in 3-space dimensions. The method is developed and analysed theoretically and experimentally. In the fifth Chapter a new method is developed, i.e. the implicit block, explicit overrelaxation (IBEB) iterative method, in which we solve the 2xl point block (or 2x2 point block) explicitly then grouping the new explicit point equations in an implicit iterative method. In this situation, two iteration parameters are used. These composite methods are analysed and some numerical experiments are carried out. In Chapter 6 we investigate the solution of one dimensional boundary value problems using the new explicit 2,3,4,6,8 and 12 point group iterative methods. Also, some non-linear boundary value problems are solved using a similar group strategy and in particular the 2,3 and 4 point group nonlinear over-relaxation method. Furthermore in a more convincing and realistic situation we examine the linear and non-linear boundary value problems using the alternating group explicit (AGE) method. Numerical results were carried out to compare these methods with the dire.ct approach using Picard's method. In a similar manner to the four and nine point groups of Chapter 4, explicit four and nine point groups to the 9 point finite difference equation are presented in Chapter 7. The methods are analysed both theoretically and experimentally. Further in this chapter theoretical results for the explicit four and nine point groups to the 13 point finite difference equation of the biharmonic operator are presented. Finally, the thesis concludes with a chapter summarizing the main results and recommendations for further work.