SIAM J. ScI. STATo COMPUT. Vol. 11, No. 4, pp. 603-620, July 1990 1990 Society for Industrial and Applied Mathematics 001 POTENTIAL FLOW IN CHANNELS* L. GREENGARDt Abstract. A method is presented for calculating potential flows in infinite channels. Given a collection of N sources in the channel and a zero normal flow boundary condition, the method requires an amount of work proportional to N to evaluate the induced velocity field at each source position. It is accurate to within machine precision and for its performance does not depend on the distribution of the sources. Like the Fast Multipole Method developed by Greengard and Rokhlin [J. Comput. Phys., 73 (1987), pp. 325-348], it is based on a recursive subdivision of space, knowledge of the governing Green’s function, and the use of asymptotic representations of the potential field. Previous schemes have been based either on conformal mapping, which experiences numerical difficulties with the domain boundary, or direct evaluation of Green’s function. Both require O(N2) work. Key words, fluid dynamics, potential flow, vortex method, N-body problem, Fast Multipole Method AMS(MOS) subject classifications. 30B50, 31A15, 41A30, 65E05, 70C05, 70F10 1. Introduction. The evaluation of potential fields in infinite channels arises as a numerical problem in several areas, most notably electrostatics and fluid dynamics. The governing equation is the Poisson equation, (1) A= subject to an appropriate boundary condition. In this paper, we will restrict our attention to two-dimensional models and will consistently use the terminology of fluid dynamics. In viscous incompressible flow, the left-hand side is the stream function, the right-hand side is the vorticity, and the condition imposed on the boundary is that of zero normal flow (2) u.n=0, where the velocity field u is given by (3) u= xx" In terms of the stream function, this is equivalent to specifying homogeneous Dirichlet boundary conditions (4) We will concentrate on particle models, where the vorticity field is discretized, not by a mesh, but by N point vortices, N (5) -- E i" l(X--Xi, Y--Yi)" i=l Here, $ is the Dirac -function and sci is the strength of the ith point vortex located at (xi, yi). In vortex methods, what we would like to compute is the stream function and/or velocity field at each particle position. In the absence of boundary effects, the * Received by the editors September 19, 1988; accepted for publication (in revised form) April 26, 1989. This work was supported in part by the Office of Naval Research under grant N00014-86-K-0310 and in part by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship. ? Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012. 603