A preconditioned p‐multigrid solution approach for the high‐order flux reconstruction method

Abstract

AbstractThis article addresses two acceleration techniques in the context of high‐order methods: p‐multigrid and local Mach number preconditioning. The flux reconstruction method is used as the spatial discretization scheme and the flow of interest is modeled by the two‐dimensional Euler and Navier–Stokes equations in both steady and unsteady settings. The Weiss and Smith low Mach number preconditioner is used together with dual time stepping in order to perform unsteady simulations. The p‐multigrid uses the third order explicit Runge–Kutta (RK3) scheme as the smoother and the non‐linear LU‐SGS implicit method as the coarse level solver. The algorithm performance is compared against both the explicit RK3 and the LU‐SGS approaches. The use of the Mach number preconditioning significantly increases the efficiency of the p‐multigrid method. For unsteady simulations, the preconditioner helps with the efficiency of the p‐multigrid with larger physical time steps. In most cases, the preconditioned p‐multigrid approach is comparable to or faster than the implicit LU‐SGS algorithm and requires less memory, specially for schemes.