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We develop a $ P $-multigrid solver to simulate locally preconditionedunsteady compressible Navier-Stokes equations at low Mach numbers with implicithigh-order methods. Specifically, the high-order flux reconstruction/correctionprocedure via reconstruction (FR/CPR) method is employed for spatialdiscretization and the high-order time integration is conducted by means of theexplicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) method.Local preconditioning is used to alleviate the stiffness of the compressibleNavier-Stokes equations at low Mach numbers and is only conducted in pseudotransient continuation to ensure the high-order accuracy of ESDIRK methods. Weemploy the element Jacobi smoother to update the solutions at different $ P$-levels in the $ P $-multigrid solver. High-order spatiotemporal accuracy ofthe new solver for low-Mach-number flow simulation is verified with theisentropic vortex propagation when the Mach (Ma) number of the free stream is0.005. The impact of the hierarchy of polynomial degrees on the convergencespeed of the $ P $-multigrid method is studied via several numericalexperiments, including two dimensional (2D) inviscid and viscous flows over aNACA0012 airfoil at $\text{Ma} = 0.001$, and a three dimensional (3D) inviscidflow over a sphere at $\text{Ma} = 0.001$. The $ P $-multigrid solver is thenapplied to coarse resolution simulation of the transitional flows over anSD7003 wing at $ 8^\circ $ angle of attack when the Reynolds number is 60000and the Mach number is 0.1 or 0.01.

An implicit $ P $-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier-Stokes equations at low Mach numbers