In honor of the sixtieth birthday of

1 Introduction This paper is written to commemorate the sixtieth birthday of Paul Garabedian. While Paul has made broad ranging contributions in mathematics, fluid dynamics and plasma physics, his work on computational aerodynamics had a very forceful impact at a critical juncture in the development of this subject. I believe that this work should be regarded as being very much in the tradition of classical applied aerodynamics. Airplanes fly by generating a well organized and carefully controlled flow over the lifting surfaces: consequently useful predictions do not necessarily require solution of full Navier Stokes equations, and can be accomplished with simplified mathematical models. The task of the applied mathematician is to identify a mathematical model which is relevant to the problem at hand and also amenable to solution. Viscous effects on full scale aircraft are essentially limited to a thin boundary layer which forms along the surface, and very little error is incurred by ignoring viscosity outside the boundary layer. Major advances were achieved in the course of the last century using the concept of an ideal fluid, both inviscid and incompressible, and this was the basis of a successful theory of airfoils and wings. As flight speeds increased to the point that the flow was dominated by compressibility effects, the focus moved to transonic flow, and it was in this field that Paul garabedian's contributions were particularly significant. Transonic flow is important because it covers an efficient operating regime for long range aircraft. To a first approximation cruising efficiency is proportional to ML D where M is the Mach number (speed divided by the speed of sound), L is the lift, and D is the drag. As long as the speed is well below the speed of sound, the lift to drag ratio does 2 not vary much with speed, so it pays to increase the speed until the effects of compressibility start to cause a radical change in the flow. This occurs when embedded pockets of supersonic flow appear, generally terminating in shock waves. A typical transonic flow pattern over a wing is illustrated in Figure 1. As the Mach number is increased the shock waves become strong enough to cause a sharp increase in drag, and finally the pressure rise through the shock waves becomes so large that the boundary layer separates. An unsteady buffeting flow may then develop, leading to violent vibrations which cannot be tolerated …