220 research outputs found
Explicit alternating direction methods for problems in fluid dynamics
Recently an iterative method was formulated employing a new splitting strategy for the
solution of tridiagonal systems of difference equations. The method was successful in solving the systems of equations arising from one dimensional initial boundary value problems,
and a theoretical analysis for proving the convergence of the method for systems whose
constituent matrices are positive definite was presented by Evans and Sahimi [22]. The
method was known as the Alternating Group Explicit (AGE) method and is referred to
as AGE-1D. The explicit nature of the method meant that its implementation on parallel
machines can be very promising.
The method was also extended to solve systems arising from two and three dimensional
initial-boundary value problems, but the AGE-2D and AGE-3D algorithms proved to be
too demanding in computational cost which largely reduces the advantages of its parallel
nature.
In this thesis, further theoretical analyses and experimental studies are pursued to establish
the convergence and suitability of the AGE-1D method to a wider class of systems arising
from univariate and multivariate differential equations with symmetric and non symmetric
difference operators. Also the possibility of a Chebyshev acceleration of the AGE-1D
algorithm is considered.
For two and three dimensional problems it is proposed to couple the use of the AGE-1D
algorithm with an ADI scheme or an ADI iterative method in what is called the Explicit
Alternating Direction (EAD) method. It is then shown through experimental results that
the EAD method retains the parallel features of the AGE method and moreover leads to
savings of up to 83 % in the computational cost for solving some of the model problems.
The thesis also includes applications of the AGE-1D algorithm and the EAD method to
solve some problems of fluid dynamics such as the linearized Shallow Water equations,
and the Navier Stokes' equations for the flow in an idealized one dimensional Planetary
Boundary Layer.
The thesis terminates with conclusions and suggestions for further work together with a
comprehensive bibliography and an appendix containing some selected programs
The spline approach to the numerical solution of parabolic partial differential equations
This thesis is concerned with the Numerical Solution of Partial
Differential Equations.
Initially some definitions and mathematical background are given,
accompanied by the basic theories of solving linear systems and other
related topics. Also, an introduction to splines, particularly cubic
splines and their identities are presented. The methods used to solve
parabolic partial differential equations are surveyed and classified
into explicit or implicit (direct and iterative) methods. We
concentrate on the Alternating Direction Implicit (ADI), the Group
Explicit (GE) and the Crank-Nicolson (C-N) methods.
A new method, the Splines Group Explicit Iterative Method is
derived, and a theoretical analysis is given. An optimum single
parameter is found for a special case. Two criteria for the
acceleration parameters are considered; they are the Peaceman-Rachford
and the Wachspress criteria. The method is tested for different
numbers of both parameters. The method is also tested using single
parameters, i. e. when used as a direct method. The numerical results
and the computational complexity analysis are compared with other
methods, and are shown to be competitive. The method is shown to have
good stability property and achieves high accuracy in the numerical
results.
Another direct explicit method is developed from cubic splines;
the splines Group Explicit Method which includes a parameter that can
be chosen to give optimum results. Some analysis and the computational
complexity of the method is given, with some numerical results shown
to confirm the efficiency and compatibility of the method.
Extensions to two dimensional parabolic problems are given in a
further chapter.
In this thesis the Dirichlet, the Neumann and the periodic
boundary conditions for linear parabolic partial differential equations
are considered.
The thesis concludes with some conclusions and suggestions for
further work
Parallel algorithms for the solution of elliptic and parabolic problems on transputer networks
This thesis is a study of the implementation of parallel algorithms for solving
elliptic and parabolic partial differential equations on a network of transputers.
The thesis commences with a general introduction to parallel processing. Here a
discussion of the various ways of introducing parallelism in computer systems and the
classification of parallel architectures is presented.
In chapter 2, the transputer architecture and the associated language OCCAM are
described. The transputer development system (TDS) is also described as well as a
short account of other transputer programming languages. Also, a brief description of
the methodologies for programming transputer networks is given. The chapter is
concluded by a detailed description of the hardware used for the research. [Continues.
Analytical and Numerical Methods for Differential Equations and Applications
The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic
The numerical solution of sparse matrix equations by fast methods and associated computational techniques
The numerical solution of sparse matrix equations by fast methods and associated computational technique
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