The Theory of Fast Projection Methods for High-Fidelity Fast Solution of the Navier-Stokes Equations

Fast projection methods are high-temporal order techniques that enable compuations of the Navier-Stokes equations at a considerable cost reduction by skipping the solution of the pressure. First coined by Capuano et al. the methods saw a revival circa 2016 and have since seen renewed interest. However, skipping a projection will result in loss of order and deterioration of absolute stability. A judicious choice of a pressure approximation must replace the solution of the pressure. This choice of a pressure appoximation has been central to recent developments in fast projection methods as it must retain accuracy and stability. Historically, this choice has been somewhat ad-hoc. In this article, we present a generalized mathematical framework to analyze and develop fast projection methods for explicit multistage Runge-Kutta integrators. The approach is based on using rooted trees to analyze order of accuracy and subsequently derive suitable analytical forms for the solution of the pressure field. Those forms are then approximated to adequate order of accuracy so that the resulting methods do not violate formal order of accuracy. In the process, several families of pressure approximations can be derived whose parametric solutions can be further optimized for stability and other RK integrator properties. We test our theoretical results for cases with periodic, stationary, and time dependent boundary conditions such as the Taylor vortex, lid-driven cavity, and channel flow.