Numerical Applications of Reflection to Partial Differential Equations

Recent papers have reported results on the numerical solution of nonlinear free boundary problems wherein a conformai transformation (which must be determined) maps the unknown flow region onto a known domain; the boundary conditions are handled by the method of steepest descent. The present paper discusses the use of the reflection property of solutions of elliptic equations to determine these boundary conditions. The procedure is applied to the vena contracta models, and it is seen that it converges about ten times faster than the steepest-descent method. This paper will report the results of a study of the application of the procedure of analytic continuation to the numerical solution of partial differential equations, and in particular, to the vena contracta problem. The possibility of extending the solution of an analytic boundary value problem across the boundary has been explored in [1], [2], and [3]. The results show that in many cases (including nonlinear problems), there is a formula expressing the solution at points beyond the boundary in terms of the values in the original domain; such an expression is commonly called a reflection rule. Here we shall examine in detail the theory behind the reflection scheme for the problem of finding a conformai transformation with certain boundary restrictions. This concept has an immediate application in the numerical solution of free boundary problems, such as the vena contracta [4], [5]. The conformai map is used to transform the complicated, unknown domain of the solution to a simpler, known domain, wherein the (transformed) differential equation can be more easily solved; of course, one must solve for the transformation simultaneously. The result is then regarded as a parametrized form of the solution. The principal difficulty in this procedure is the handling of the boundary conditions for the conformai map. In [4] Bloch derived boundary equations from a steepest-descent argument, supplemented by equations serving to establish the free boundary constraint (which in the case of the vena contracta expresses continuity of the pressure across a fluid-air interface). Our paper purports to demonstrate that the reflection technique is superior to the method of steepest descent in handling these problems. Part 1 of this paper describes the theory behind this reflection scheme. Proofs of certain results are given when available, and numerical evidence of other consequences Received May 18, 1972; revised June 18, 1975. AMS (MOS) subject classifications (1970). Primary 30A28, 35-04; Secondary 65M05, 76B10.