POWER SERIES SOLUTION, BY MACHINE, OF A NONLINEAR PROBLEM IN TWO‐DIMENSIONAL FLUID FLOW*

Abstract

SummaryThe solution of a partial differential equation, with Cauchy data given on an initial surface, can be obtained as a power series expansion in the independent variables about a point on the surface if the functions appearing in the differential equations, the equation for the initial surface, and the Cauchy data are all analytic, as has been known since the work of Cauchy and Kowalewski. This power series and its analytic continuations can be used in principle for calculating the solution. The practical application of these ideas to a certain class of problems, with the use of a modern computing machine, can be achieved by means of techniques that have been developed at New York University. Typical of the class is the problem of determining the flow of air past an axially symmetric blunt body moving at supersonic speed. This flow contains a detached shock ahead of the body and, aft of the shock, a subsonic region near the nose and a supersonic region farther downstream. Normally heat conduction and viscosity can be ignored, except in a boundary layer of negligible thickness on the surface of the body. If the problem is inverted–if the shock is assumed to lie on a given analytic surface and the Rankine‐Hugoniot jump conditions at the shock are used to provide Cauchy data from which the flow aft of the shock can be obtained, and from which, in turn, the body shape required to produce the assumed shock and resulting flow can be determined, there results a problem of the kind described, with a fourth‐order system of nonlinear partial differential equations in two independent variables. The techniques described, which yield a powerful computational procedure for problems of this kind, are (1) algorithms that enable the machine to perform the needed algebraic as well as the numerical work; (2) special floating‐point subroutines, utilizing the so‐called significance arithmetic; (3) methods for obtaining analytic extensions of the functions describing the flow. These techniques and typical results are described.