A fast Cauchy-Riemann solver

We present a solution algorithm for a second-order accurate discrete form of the inhomogeneous Cauchy-Riemann equations. The algorithm is comparable in speed and storage requirements with fast Poisson solvers. Error estimates for the dis- crete approximation of sufficiently smooth solutions of the problem are established; numerical results indicate that second-order accuracy obtains even for solutions which do not have the required smoothness. Different combinations of boundary conditions are considered and suitable modifications of the solution algorithm are described and implemented. 1. Introduction. Inhomogeneous Cauchy-Riemann equations appear naturally in many fluid-dynamical problems, as the divergence and the vorticity equations of a two-dimensional steady flow field (u, v) = (t/(x, y), u(x, y)). The velocity components u, v axe usually called in this context primitive variables, in contradistinction to the derived variables \ii, f in the stream function-vorticity formulation of the flow equa- tions (e.g., Roache (28)). In the latter formulation, the stream function \p satisfies a Poisson equation; and computations with this formulation have greatly benefited from the rapid development of fast direct methods for the solution of Poisson's equation, or Poisson solvers (Buneman (4), Buzbee, Golub and Nielson (5), Dorr (9), Fischer, Golub, Hald, Leiva and Widlund (12), Golub (18), Hockney (21), (22), Widlund (32) ). Working in the primitive variables, however, permits the treatment of more gen- eral flows. Indeed, either nondivergence or irrotationality of the flow are required in order to introduce a stream function \p or a velocity potential 0, and obtain a Poisson equation for them. There are many situations of practical interest in which neither of these assumptions holds. Furthermore, the formulation of boundary conditions is often easier in terms of the primitive variables, by using physical considerations which arise naturally from the problem. On the other hand, a boundary condition on the vorticity f for instance is at times hard to formulate (Langlois (23)); the construction of appropriate discrete versions of such a boundary condition is often even more diffi- cult (Oliger and Sundstrom (27)). Hence, the desirability of simple, physically mean- ingful boundary conditions and, thus, of the use of primitive variables. Lomax and Martin (24) have developed a fast Cauchy-Riemann solver and