An overview and generalization of implicit Navier-Stokes algorithms and approximate factorization

A theme of linearization and approximate factorization provides the context for a retrospective overview of the development and evolution of implicit numerical methods for the compressible and incompressible Euler and Navier- Stokes algorithms. This topic was chosen for this special volume commemorat ing the recent retirements of R.M. Beam and R.F. Warming. A generalized tre atment of approximate factorization schemes is given, based on an operator notation for the spatial approximation. The generalization focuses on the i mplicit structure of Euler and Navier-Stokes algorithms as nonlinear system s of partial differential equations, with details of the spatial approximat ion left to operator definitions. This provides a unified context for discu ssing noniterative and iterative time-linearized schemes, and Newton iterat ion for unsteady nonlinear schemes. The factorizations include alternating direction implicit, LU and line relaxation schemes with either upwind or ce ntered spatial approximations for both compressible and incompressible flow s. The noniterative schemes are best suited for steady flows, while the ite rative schemes are well suited for either steady or unsteady flows. This ge neralization serves to unify a large number of schemes developed over the p ast 30 years. (C) 2001 Elsevier Science Ltd. All rights reserved.