Reflections on the evolution of implicit Navier–Stokes algorithms

This article gives a historical perspective on the evolution of implicit algorithms for the Navier–Stokes equations that utilize time or Newton linearizations and various forms of approximate factorization including alternating-direction, lower/upper, and symmetric relaxation schemes. A theme of the paper is how progress in implicit Navier–Stokes algorithms has been influenced and enabled by the introduction of characteristic-based upwind approximations, unstructured-grid discretizations, parallelization, and by advances in computer performance and architecture. Historical examples of runtime, problem size, and estimated cost are given for actual and hypothetical Navier–Stokes flow cases from the past 40 years.