Browsing by Author "Parkyn, D G"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Open Access
    Analytical differentiation by computer using a Symmetrical List Processor
    (1972) Lichtman, Barry Martin; Parkyn, D G
    The Symmetrical List Processor SLIP; developed by Professor Joseph Weizenbaum of MIT, was implemented with considerable modifications and additions on the University of Cape Town computer. A package to perform automated analytical differentiation (DERIV) was developed using SLIP. Basic simplification techniques as well as convenient input and output routines were included. The package was tested extensively and a rough comparison drawn with the abilities of various computer languages and programs which include the same facility as TIERIV.
  • Thumbnail Image
    Item
    Open Access
    Explicit approximation methods for initial-value problems
    (1969) Joubert, Gerhard Robert; Parkyn, D G
    Explicit difference approximations of parabolic initial boundary value problems are usually stable only if a difference grid with a limited time-step is used. By considering the one-dimensional diffusion equation as an example, it is shown in the following work that simple smoothing formulas can be constructed which, when applied to solutions computed with unstable explicit difference equations, result in stable approximations of the solution of the differential equation. Such computational procedures can be expressed as explicit difference analogues of the problem considered. Conversely, explicit difference approximations, which need not be defined for all points of the difference grid but must be stable for the specific grid used, can be written as non-unique combinations of an explicit difference approximation, which need not be stable, and a smoothing formula. By appropriate choice of these explicit difference approximations and smoothing formulas this procedure will be defined for all grid points. This new technique thus has the advantage that explicit difference approximations with comparatively weak stability requirements and/or small truncation errors can be used in practice.
  • Thumbnail Image
    Item
    Open Access
    Forced oscillations in simple and binary gas atmospheric models
    (1970) Viljoen, Michael David; Parkyn, D G
    The presence of periodic oscillations in the earth's atmosphere has been confirmed in recent years by analysis of satellite drag data. The amplitudes of these oscillations vary with height and time in a complex manner with the underlying physical mechanism of this behaviour not fully understood. Classical studies have been limited to the lower levels of the atmosphere, but these have neglected to include the damping effects of heat conduction and viscosity. These also ignored the second order terms in the equations of motion which, in effect, treats an otherwise singular perturbation problem as regular. Upper atmosphere studies of the diurnal density oscillations were discussed by Nicolet, on the basis of the mutual diffusion of the components of a binary gas system, where he compared different equilibrium configurations. This statical treatment again ignores the damping effects on mass flow. D.G. Parkyn reduced the problem in idealised form to that of investigating the effect of a travelling temperature wave at the base of a viscous, heat conducting, diffusing gas atmosphere. This model excludes molecular dissociation and ionization in the upper regions and absorption of solar energy. Incorporation of all these properties would render the problem impossibly difficult . As a first step to the development of the analysis for the complex spherical atmosphere, Parkyn simplified the model to that of a cylindrical homogeneous atmosphere, and he considered the effect of forced oscillations about an isothermal equilibrium state. Parkyn showed that this idealised problem is capable of explicit solution and, contrary to the result of Wilkes, he found that the amplitudes of the forced osciliations decrease with height in the lower atmospheric regions. This implies the importance of heat conduction and viscosity as damping effects in these regions. It is proposed to extend the analysis of Parkyn by simplifying the geometry, treating one space dimension, so giving more flexibility to the assumed physical properties. Continuum equations of motion will be taken to hold throughout the range of investigation.