Efficient p ‐multigrid discontinuous Galerkin solver for complex viscous flows on stretched grids

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.

Section: Steady Flow Simulationsmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

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

Jiang

Yan

et al. 2015

“…Work is underway to implement a dual time stepping technique to solve the nonlinear system arising at each MEBDF stage, based on a p ‐multigrid strategy , and to compare the performance of different high‐order time integration schemes (MEBDF, TIAS , ESDIRK, and Rosenbrock methods). Preliminary results for compressible inviscid and viscous test cases show that MEBDF4 scheme performs better than ESDIRK4 and TIAS4, but it is outperformed by TIAS6 and Rosenbrock (with order greater than three) schemes.…”

Section: Discussionmentioning

confidence: 99%

Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows

Nigro

Ghidoni

Rebay

et al. 2014

Self Cite

SUMMARYIn this paper, a high-order DG method coupled with a modified extended backward differentiation formulae (MEBDF) time integration scheme is proposed for the solution of unsteady compressible flows. The objective is to assess the performance and the potential of the temporal scheme and to investigate its advantages with respect to the second-order BDF. Furthermore, a strategy to adapt the time step and the order of the temporal scheme based on the local truncation error is considered. The proposed DG-MEBDF method has been evaluated for three unsteady test cases: (i) the convection of an inviscid isentropic vortex; (ii) the laminar flow around a cylinder; and (iii) the subsonic turbulent flow through a turbine cascade.

“…The objective functional, formally obtained by increasing the degree of the discretization from k to k +1, is approximated to avoid the computational expense of solving a higher degree DG discretization of the nonlinear system of equations, see also Hartmann et al and Hartmann and Houston . The approximation relies on the Taylor series expansion of J k +1 ( W k +1 ) about the L 2 ‐projection

{boldw}_{k}^{k + 1}

of the solution w k on V k +1

J_{k + 1} ((), W_{k + 1}) \approx J_{k + 1} ((), W_{k}^{k + 1}) + {((), \frac{\partial J_{k + 1}}{\partial W_{k + 1}})}_{{boldW}_{k}^{k + 1}} ((), W_{k + 1} - W_{k}^{k + 1}) .

Similarly, the vector of the residuals is expanded about

{boldW}_{k}^{k + 1}

R_{k + 1} ((), W_{k + 1}) \approx R_{k + 1} ((), W_{k}^{k + 1}) + {([], \frac{\partial R_{k + 1}}{\partial W_{k + 1}})}_{{boldW}_{k}^{k + 1}} ((), W_{k + 1} - W_{k}^{k + 1}) .

In practice, when orthogonal and hierarchical modal bases are used, the degrees of freedom of the projected solution are the same of the low‐order one with null higher‐order modes …”

Section: Adjoint‐based Error Estimationmentioning

confidence: 99%

“…In practice, when orthogonal and hierarchical modal bases are used, the degrees of freedom of the projected solution are the same of the low-order one with null higher-order modes. 36 Since the unknown W k+1 is the solution of the higher-order discretization of the nonlinear problem, we have R k+1 (W k+1 ) = 0. Thus, by rearranging Equation 14, we obtain the following error representation…”

Section: Formulation Of the Adjoint Problemmentioning

confidence: 99%

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

Zenoni

Leicht

Colombo

et al. 2017

Self Cite

Summary In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.