PARALLEL P-ADAPTIVE IMPLICIT HIGH-ORDER FLUX RECONSTRUCTION METHODS FOR UNDER-RESOLVED TURBULENT FLOW SIMULATION

Abstract

A simplified flux reconstruction method, i.e., the compact direct flux reconstruction method, is developed using the compact finite difference approach within the standard element. It can be regarded as a differentiation form of the direct flux reconstruction method. Implicit high-order time integration methods are employed to achieve high-order spatiotemporal accuracy and circumvent the restriction on the time step size. To efficiently solve the nonlinear and linear systems resulting from the high-order discretization, we employ the Newton--Krylov solver as well as the p-multigrid solver. Specifically, the matrix-free implementation of the generalized minimal residual method is employed to significantly reduce memory consumption. The element-Jacobi preconditioner is used for the generalized minimal residual method and it also serves as a smoother for the p-multigrid solver. The impact of the polynomial hierarchy on the convergence speed of the V-cycle p-multigrid solver is discussed to reveal that the polynomial difference between two adjacent levels should be half of the polynomial degree on the finer level. The local preconditioning technique is employed to solve the Navier--Stokes equations at low Mach numbers. The local preconditioning can preserve the accuracy of high-order methods when the Mach number is small and accelerate the convergence of implicit methods. To further increase the efficiency of high-order methods when solving massive turbulent flows, a dynamic p-adaptation method is developed with a dynamic load balancing technique. When the p-adaptive implicit high-order method is applied to under-resolved turbulence simulation, it can significantly decrease the total number of solution points (up to 76%) as well as the run time (up to 70%). Overall, in this work, implicit high-order numerical methods with various acceleration techniques, namely, implicit time stepping, local preconditioning, p-multigrid solver, and p-adaptation, are developed and studied as our endeavor towards the efficient, accurate, and robust simulation of turbulent flows, including low-Mach-number problems.