Practical aspects of p‐multigrid discontinuous Galerkin solver for steady and unsteady RANS simulations

Abstract

SUMMARYEfficient and robust p-multigrid solvers are presented for solving the system arising from high-order discontinuous Galerkin discretizations of the compressible Reynolds-Averaged Navier-Stokes (RANS) equations. Two types of multigrid methods and a multigrid preconditioned Newton-Krylov method are investigated, and both steady and unsteady algorithms are considered in this paper. For steady algorithms, a new strategy is introduced to determine the CFL number, which has been proved to be critical in achieving the effective and stable convergence for p-multigrid methods. We also suggest a modified smoothing technique to further improve the efficiency of the algorithms. For unsteady algorithms, special attention has been paid to the cycling strategy and the full multigrid technique, and we point out a significant difference on the parameter selection for unsteady computations. The capabilities of the resulted solvers have been examined by performing steady and unsteady RANS simulations. Comparative assessment in terms of efficiency, robustness, and memory consumption are carried out for all solvers.

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

“…The factorization cost of element‐wise implicit solvers for 3‐D simulations grows rapidly with the polynomial order, proportional to $$$ {\left(p+1\right)}^9 $$$, as well as the memory footprint, proportional to $$$ {\left(p+1\right)}^6 $$$. A lot of developments have already been made in those two areas, namely, the p‐multigrid method 12‐22 and the Mach number preconditioning technique 23‐28 . However, it does not seem that an attempt has been made to combine both p‐multigrid and preconditioning techniques in an unsteady high‐order framework with an explicit iterative smoother.…”