ible codes in terms of both convergence rate and accuracy suffers greatly. In Ref. [54], Volpe demonstrated the poor An implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the performance of compressible flow codes under these condicode is a backward Euler time discretization for which the linear tions, particularly for Mach numbers below approxisystem is approximately solved at each time step with either a point mately 0.1.
implicit method or a preconditioned generalized minimal residual
To overcome the difficulties associated with use of com-(GMRES) technique. For the GMRES calculations, several techpressible codes, excellent progress has been made in the niques are investigated for forming the matrix-vector product. Convergence acceleration is achieved through a multigrid scheme that use of local preconditioners to extend the applicability of uses nonnested coarse grids that are generated using a technique these codes to low Mach numbers. Several examples of described in the present paper. Convergence characteristics are this technique, as well as the necessary theory, can be investigated and results are compared with an exact solution for found in Refs. [14, 19,25,49,50,51, 56]. Preconditioning the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow is indeed a viable means of extending the applicability of over a NACA 4412 airfoil, a three-element airfoil for which Mach compressible flow codes to the low-Mach-number range number effects are investigated, and three-dimensional flow over