2134/12034
Wadalla S. Yousif
New block iterative methods for the numerical solution of boundary value problems
2013
Loughborough University
untagged
2013-04-03 12:40:19
article
https://repository.lboro.ac.uk/articles/New_block_iterative_methods_for_the_numerical_solution_of_boundary_value_problems/9405986
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.