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 commemorating the recent retirements of R.M. Beam and R.F. Warming. A generalized treatment of approximate factorization schemes is given, based on an operator notation for the spatial approximation. The generalization focuses on the implicit structure of Euler and Navier–Stokes algorithms as nonlinear systems of partial differential equations, with details of the spatial approximation left to operator definitions. This provides a unified context for discussing noniterative and iterative time-linearized schemes, and Newton iteration for unsteady nonlinear schemes. The factorizations include alternating direction implicit, LU and line relaxation schemes with either upwind or centered spatial approximations for both compressible and incompressible flows. The noniterative schemes are best suited for steady flows, while the iterative schemes are well suited for either steady or unsteady flows. This generalization serves to unify a large number of schemes developed over the past 30 years.