Engineering Transactions, 9, 1, pp. 5-18, 1961

Numeryczne Modelowanie Procesów Fizycznych

J.R.M. Radok
Politechnic Institute of Brooklin
United States

Many physical processes are most conventiently described by partial differential equations relating the rates of change of the dependent variables to their space distributions. Occasionally, these relationships may also involve integrals over the past histories of these variables. The solutions of problems, governed by such equations, become particular by the enforcement of given initial and boundary conditions the number of which depends on the equations themselves. Prior to the advent of high speed computers, and, as a consequence, of almost infinite numbers of arithmetic operations, a specific problem was considered to have been solved exactly, if its solution had been expressed in terms of special functions which preferably had to be tabulated. As the domain of applied mathematics expanded and practical problems became more involved, their numerical aspect-received more attention, but all the time the status of numerical work was sub-sidiary to that of analysis. The numerical analyst had very little influence on the formulation of the problems requiring solution, he simply thought in terms of difference equations and systems of linear algebraic equations the number of which he had to limit in order to stay within the humanly realistic range of arithmetic operations.
So far, the modern computer seems to have caused less change in this situation than these considerations would lead one to expect. In particular, the numerical methods for solving problems have changed little from the special function days. Thus, there arise the following basic questions:
a) Are the methods designed for small numbers of arithmetic operations and maximum utilization of tabulated results suitable for the computers?
b) What criteria can be used for the development of numerical methods under the new conditions? These ideas are discussed in detail and certain aspects of the use of exponential functions as the base for asymptotic integration methods are discussed in the light of applications to problems of diffusion and beam theory. The selected approach stresses the numerical simulation aspect in order to demonstrate how physical insight and knowledge of the mathematical features of the sought solutions can be combined for the purpose of the design of numerical methods.

Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


. J.R.M. RADOK and G.D. GALLETLY, On the Accuracy of Some Shell Solutions, ASME Paper No. 59-APM-30. Presented at the 22nd Applied Mechanics Division National Conference at Rensselaer Polytechnic Institute, Troy, N.Y., June 1959.

. L. HOPF, Einführung in die Differentialgleichungen der Physik, Sammlung Göschen, Vol. 1070.

. Robert D. RICHTMYER, Difference Methods for Initial Value Problems, Interscience Publishers Inc., N.Y., New York 1957.

. J.R.M. RADOK, Method of Functional Extrapolation for the Numerical Integration of Ordinary Differential Equations, J. Soc. Ind. Appl. Math., December 1959.

. M. MUSKAT, Physical Principles of Oul Production, McGraw-Hill Book Co., New York 1949.

. R. COURANT, K.O. FRIEDRICHS and H. LEVY, Mathematische Annalen, 100, 32-74 1928; Über die Partiellen Differentialgleichungen der Mathematischen Physik.

. J. FRITZ, On Integration of Parabolic Equations by Difference Methods, Comm, Pure Appl, Math., Vol. 5, 1952, 155-211.