Preconditioned smoothers for the full approximation scheme for the RANS equations

TL;DR

This paper introduces new preconditioned smoothers for multigrid methods applied to RANS equations, demonstrating improved convergence rates and efficiency in steady and unsteady flow simulations.

Contribution

It derives and compares two classes of preconditioned smoothers, ARK and additive W, based on Rosenbrock smoothers, and evaluates their performance with SGS preconditioners.

Findings

AW3 smoother achieved the best efficiency.

Steady-state convergence rate of 0.85 with 6 orders of magnitude reduction.

Unsteady convergence rate of 0.77 with 11 orders of magnitude reduction.

Abstract

We consider multigrid methods for finite volume discretizations of the Reynolds Averaged Navier-Stokes (RANS) equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge-Kutta (ARK) methods. Furthermore, we derive the new class of additive W (AW) methods from Rosenbrock smoothers. This gives rise to two classes of preconditioned smoothers, preconditioned ARK and additive W (AW), which are implemented the exact same way, but have different parameters and properties. The new derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. Numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations.