In this study we generate optimal Runge-Kutta (RK) schemes for
converging the Artificial Compressibility Method (ACM) using dual
time-stepping with high-order unstructured spatial discretizations. We
present optimal RK schemes with between s = 2 and s = 7 stages for
Spectral Difference (SD) and Discontinuous Galerkin (DG) discretizations obtained using the Flux Reconstruction (FR) approach with
solution polynomial degrees of k = 1 to k = 8. These schemes are optimal in the context of linear advection with predicted speedup factors in
excess of 1.80× relative to a classical RK4,4 scheme. Speedup factors of
between 1.89× and 2.11× are then observed for incompressible Implicit
Large Eddy Simulation (ILES) of turbulent flow over an SD7003 airfoil.
Finally, we demonstrate the utility of the schemes for incompressible
ILES of a turbulent jet, achieving good agreement with experimental data. The results demonstrate that the optimized RK schemes
are suitable for simulating turbulent flows and can achieve significant
speedup factors when converging the ACM using dual time-stepping
with high-order unstructured spatial discretizations.