One of the mo,;t common and useful methods employed in the numerical integration of partial differential equations invoh'es the replacement of the differential equation hy an equivalent difference equation. This technique has become particularly important in rec'ent years because of the development of modern high-speed computing machines.In the present paper ,ye shall show that the accuracy of a finite difference solution to a partial differential problem is conveniently discussed in terms of the "convergence" and "stability" of the difference scheme. Courant, Friedrichs, and Lewy (reference 1) di,;cussed the conyergence of difference solutions for the basic types of linear partial differential equations; for equations of parabolic or hyperbolic character, they found the important result that the "me,;h-ratio" must satisfy certain ine~ualities . .T. von Neumann obtained the same inequalities from a study of error-growth (stability of the difference scheme). The partly heuristic technique of stability analysis developed by von Neumann was applied hy him to a wide variety of difference and differential equation problems during World War II. This method has been very briefly mentioned in the literature (references 2, 3, 4) but a detailed discussion has not yet been published. 'Vith the kind permission of Professor yon Neumann, we have made such a discussion part of the present paper.We begin with terminology and definitions. Let D represent the exact solution of the partial diffarntial equation, ~ represent the exact solution of the partial difference equation, and N represent the numerical solution of the partial difference equation. We call (D -~) the truncation error; it arises because of the finite distance between points of the difference mesh. To find the conditions under which ~ -> D is the problem of com'ergencc. We call (~ -N) the numerical error. If a faultless computer working to an infinite number of decimal places were employed, the numerical error would be zero. Although (~ -N) may consist of several kinds of errors, we usually consider it limited to roundoff errors. To find the conditions under which (~ -N) is small throughout the entire region of integration is the problem of stability.'Vhether a given finite-difference scheme tlatisfies the criteria for convergence and stability (we say, for short, that the difference-scheme is convergent/ divergent and stable/unstable) depends upon the form of the ~-equation and upon the initial and boundary conditions. If the ~-equation is linear, stability (and usually convergence al,;o) will not depend on the initial and boundary conditions. Now for most problems, D and ~ are either unavailable or can only be obtained with much greater labor than is involved in finding N. The prin-

No abstract

This paper is the case history of an operations-research study of a barge line undertaken in the summer of 1954. One problem considered was the scheduling of tugs and the determination of the resulting number of barge loads per year which would result. The second problem was to determine the proper balance between tugs and barges for three or four tugs. These problems were solved by simulating the operation, the results of the simulation led to recommendations and predictions. These predictions are compared to the actual data over the ensuing years as the recommendations were carried out.

No abstract

No abstract