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Implicit methods for hyperbolic equations are analyzed using LU decompositions. It is shown that the inversion of the resulting tridiagonal matrices is usually stable even when diagonal dominance is lost. Furthermore, these decompositions can be used to construct stable algorithms in multidimensions. When marching to a steady state, the solution is independent of the time. Alternating direction methods which solve for u n+1 − u n are unconditionally unstable in three-space dimensions and so the new method is more appropriate. Furthermore, only two factors are required even in threespace dimensions and the operation count per time step is low. Acceleration to a steady state is analyzed, and it is shown that the fully implicit method with large time steps approximates a Newton-Raphson iteration procedure.

This paper surveys the evolution of computational methods in aerodynamics. Improvements in high-speed electronic computers have made it feasible to attempt numerical calculations of progressively more complex mathematical models of aerodynamic flows. Numerical approximation methods for a hierarchy of models are examined in ascending order of complexity, ranging from the linearized potential flow equation to the Reynolds averaged Navier Stokes equations, with the inclusion of some previously unpublished material on implicit and multigrid methods for the Euler equations. It is concluded that the solution to the Euler equations for inviscid flow past a complete aircraft is a presently attainable objective, while the solution to the Reynolds averaged Navier Stokes equations is a possibility clearly visible on the horizon.

The method he5 been succ e s s f u i l y a p p l i e d t o two-dimensional a i r f o i l c a l c ul a t i o n s on b o t h & t y p e and C-type meshes. I n t h r e e d i m e n s i o n s t h e scheme has proved e q u a l l y e f f e c t i v e and c a l c u l a t i o n s of f l o w s over wing/body cornb i n a t i o n s a r e p o s s i b l e w i t h conve-gence achieved i n l e s s t h a n 100 cycles.

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