Sahimi, Mohd S.
Numerical methods for solving hyperbolic and parabolic partial differential equations
The main object of this thesis is a study of the numerical
'solution of hyperbolic and parabolic partial differential equations.
The introductory chapter deals with a general description and classification
of partial differential equations. Some useful mathematical
preliminaries and properties of matrices are outlined.
Chapters Two and Three are concerned with a general survey of
current numerical methods to solve these equations. By employing
finite differences, the differential system is replaced by a large
matrix system. Important concepts such as convergence, consistency,
stability and accuracy are discussed with some detail. The group explicit (GE) methods as developed by Evans and Abdullah
on parabolic equations are now applied to first and second order (wave
equation) hyperbolic equations in Chapter 4. By coupling existing
difference equations to approximate the given hyperbolic equations, new
GE schemes are introduced. Their accuracies and truncation errors are
studied and their stabilities established.
Chapter 5 deals with the application of the GE techniques on some
commonly occurring examples possessing variable coefficients such as
the parabolic diffusion equations with cylindrical and spherical
symmetry. A complicated stability analysis is also carried out to
verify the stability, consistency and convergence of the proposed scheme.
In Chapter 6 a new iterative alternating group explicit (AGE)
method with the fractional splitting strategy is proposed to solve
various linear and non-linear hyperbolic and parabolic problems in one
dimension. The AGE algorithm with its PR (Peaceman Rachford) and DR (Douglas Rachford) variants is implemented on tridiagonal systems of
difference schemes and proved to be stable. Its rate of convergence
is governed by the acceleration parameter and with an optimum choice
of this parameter, it is found that the accuracy of this method, in
general, is better if not comparable to that of the GE class of problems
as well as other existing schemes.
The work on the AGE algorithm is extended to parabolic problems of
two and three space dimensions in Chapter 7. A number of examples are
treated and the DR variant is used because of consideration of stability
requirement. The thesis ends with a summary and recommendations for
future work.
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2013-04-09
https://repository.lboro.ac.uk/articles/Numerical_methods_for_solving_hyperbolic_and_parabolic_partial_differential_equations/9406871