| dc.description.abstract | This thesis is based on the investigations carried out by the author on the numerical solution of first-order systems of hyperbolic equations. The thesis has been organized into four chapters. The first chapter is introductory and presents a brief survey of the relevant earlier work, the motivation behind the study, and a summary of the results reported in the thesis.
The method of characteristics for solving first-order systems of hyperbolic equations offers several advantages compared to other methods. Firstly, the original equations governing the physical phenomenon are simplified on the characteristic surfaces, and the number of derivatives is reduced. The region of influence is determined rigorously, and thus the propagation of perturbations is properly taken into account. This gives a clear physical sense to the method of characteristics and enables one to rigorously consider certain important features of the physical phenomenon.
In the second chapter, a general second-order method—of which the most widely used Butler’s method [1] is a particular case—is developed for the first-order system of three equations equivalent to the wave equation in two space dimensions. Two examples—an initial boundary value problem and a pure initial value problem—are solved to compare the results of two particular cases of the general method and Butler’s method, and to show the accuracy and stability of the schemes. The problems are also solved using the locally one-dimensional or splitting method of Strang [2] to demonstrate the superiority of the second-order bicharacteristic methods over second-order finite-difference methods.
Though the bicharacteristic methods are more accurate, they require more computational effort than finite-difference methods. Moreover, it is difficult to extend bicharacteristic methods to higher orders of accuracy and to higher dimensions, whereas the same is straightforward with finite-difference methods. This is one of the reasons why finite-difference methods are still widely used. Recently, higher-order methods have caught the attention of researchers in this field.
In Part A of Chapter III, we propose an explicit algorithm which generates
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order accurate explicit methods when a
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order accurate method and a
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order accurate method are known. Using this algorithm, one can generate a method of desired accuracy in any number of space dimensions. As examples, a third-order scheme and a fourth-order scheme in one space dimension are given, and the stability criteria have been derived for them. The numerical results of a few test problems are presented to show the accuracy and stability of the schemes.
The study of the stability of the numerical solution of initial boundary value problems using explicit higher-order methods in the interior of the region is quite involved, as explicit methods require a larger stencil depending on the order of the method. Implicit methods usually require a stencil with considerably fewer mesh points. In Part B of Chapter III, generalizations of leap-frog methods, which give fourth-order accuracy, are studied. Stability criteria of these generalizations for initial value problems are established. Phase errors are analyzed to show the superiority of fourth-order implicit methods over explicit methods. Numerical results, obtained by solving certain test problems in one space dimension, are presented to show accuracy and stability. The usefulness of Shuman filters, while dealing with problems having shock solutions, has also been demonstrated.
While solving partial differential equations by finite-difference approximations, the numerical solution often develops large gradients or discontinuities in a localized region. One would like to use a refined mesh in that region by employing the same finite-difference approximation or to use a different difference approximation locally to suit the phenomenon [3]. When one uses a different difference approximation to handle the local phenomenon, one would like to know whether the matching of these approximations would result in a stable solution or not. Chapter IV gives a stability criterion for the matching of general schemes. Certain particular cases have been discussed, and some specific matchings of well-known difference approximations are provided to illustrate the theory. The stability theory of the matching technique is also found useful to determine the stability of difference approximations to initial boundary value problems in the case of translatory boundary conditions.
The work reported in Part A of Chapter III has been accepted for publication [4]. Some parts of the other reported work have been communicated for publication, and the remaining will be communicated in due course. | |
