  A fully implicit high-order preconditioned flux reconstruction/correction
procedure via reconstruction (FR/CPR) method is developed to solve the
compressible Navier-Stokes equations at low Mach numbers. A dual-time stepping
approach with the second-order backward differentiation formula (BDF2) is
employed to ensure temporal accuracy for unsteady flow simulation. When dynamic
meshes are used to handle moving/deforming domains, the geometric conservation
law (GCL) is implicitly enforced to eliminate errors due to the resolution
discrepancy between BDF2 and the spatial FR/CPR discretization. The large
linear system resulted from the spatial and temporal discretizations is tackled
with the restarted Generalized Minimal Residual (GMRES) solver in the PETSc
(Portable, Extensible Toolkit for Scientific Computation) library. Through
several benchmark steady and unsteady numerical tests, the preconditioned
FR/CPR methods have demonstrated good convergence and accuracy for simulating
flows at low Mach numbers. The new flow solver is then used to study the
effects of Mach number on unsteady force generation over a plunging airfoil
when operating in low-Mach-number flows. It is observed that weak
compressibility has a significant impact on thrust generation but a negligible
effect on lift generation of an oscillating airfoil.
