On the theory of fast projection methods for high-order Navier-Stokes solvers

Abstract

We propose a new framework for the design of fast-projection solvers for the Navier-Stokes equations using Runge-Kutta integrators. The framework uses the full nonlinear equations along with rooted trees and exposes the need to track non-linear advection terms for fourth-order and higher integrators as those introduce irreducible elementary derivatives in the truncation error. The framework is then used to derive approximations to the pressure projection by replacing those with suitable approximations that do not affect formal order of accuracy. The proposed pseudo-pressure approximations are easy-to-implement with some of the variants including parametric coefficients that can be exploited to optimize fast-projection solvers for stability or other desirable properties. In addition, we report approximations for adaptive timestepping. Both the parametric and adaptive timestepping results are being reported for the first time in the literature. The proposed approximations are verified with computations of the full Navier-Stokes equations on two benchmark problems with steady and unsteady boundary conditions and fixed and adaptive timestepping. All calculations agree with the theoretical results and produce the correct formal order of accuracy. The proposed framework lays the foundation for future work on fast-projection methods and can be used to reproduce existing results in the literature.