Numerical realization of an iterative method for boundary-layer equations

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TWO modifications of the analytic method of successive approximation for solving the boundary-layer system of equations, which enable the numerical iterative process to be continued without limit on a computer, are explained. In this paper we describe a numerical realization of the method of successive approximations for solving the boundary-layer equations, explained previously in [1, 2]. Various iterative methods have been applied repeatedly to solve the boundary-layer equations and to prove existence theorems of the solution of such equations (see [3, 4]). The method of successive approximations considered is intended for solving the systems of equations of the laminar multicomponent boundary layer with an arbitrary pressure gradient on plane or axisymmetric bodies (possibly, taking into account chemical reactions). It defines a uniform, infinitely continuable iterative process of the form U j+1 = T( U j ), which in principle enables any approximation U 1, U 2, … to be obtained in analytic form. The method of [1, 2] contains some ideas formally similar to Targi and Shvets' approximate methods of calculating the boundary layer [5, 6]. These method, using the idea of a layer of finite thickness, construct only the first approximation, and the question of finding successive approximations is not discussed in these papers. Theoretically, successive approximations cannot be calculated, since they lead to differential equations of high order (for the thickness of the boundary layer) and require non-obvious supplementary initial conditions. For sufficiently complex problems, it is possible by the method described in [1, 2] to obtain analytically two or three approximations of high accuracy. No theoretical difficulties arise in the calculation of subsequent approximations, but the volume of technical work increases strongly. It would be considerably easier to obtain distant approximations by computer. Moreover, the computer realization and analysis of a large number of approximations may clarify the question of the convergence of the method. In section 1 the idea of the method is demonstrated by the example of a non-selfsimilar equation. Then, also by simple examples, in sections 2 and 3 two modifications of the analytic method of [1, 2], useful for the numerical realization of this method for problems of the type indicated above, are explained. Section 4, the last, contains some recommendations on the individual stages of the computer realization of the method, and also a brief description of the results of the solution of some model problems.