A Guide to the Implementation of Boundary Conditions in Compact High-Order Methods for Compressible Aerodynamics (original) (raw)
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International Journal for Numerical Methods in Fluids
The present paper addresses the numerical solution of turbulent flows with high-order discontinuous Galerkin methods for discretizing the incompressible Navier-Stokes equations. The efficiency of high-order methods when applied to under-resolved problems is an open issue in literature. This topic is carefully investigated in the present work by the example of the 3D Taylor-Green vortex problem. Our implementation is based on a generic high-performance framework for matrix-free evaluation of finite element operators with one of the best realizations currently known. We present a methodology to systematically analyze the efficiency of the incompressible Navier-Stokes solver for high polynomial degrees. Due to the absence of optimal rates of convergence in the under-resolved regime, our results reveal that demonstrating improved efficiency of highorder methods is a challenging task and that optimal computational complexity of solvers, preconditioners, and matrix-free implementations are necessary ingredients to achieve the goal of better solution quality at the same computational costs already for a geometrically simple problem such as the Taylor-Green vortex. Although the analysis is performed for a Cartesian geometry, our approach is generic and can be applied to arbitrary geometries. We present excellent performance numbers on modern, cache-based computer architectures achieving a throughput for operator evaluation of 3 • 10 8 up to 1 • 10 9 DoFs/sec on one Intel Haswell node with 28 cores. Compared to performance results published within the last 5 years for highorder DG discretizations of the compressible Navier-Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.
High-Order Discontinuous Galerkin Methods using a Spectral Multigrid Approach
43rd AIAA Aerospace Sciences Meeting and Exhibit, 2005
The goal of this paper is to investigate and develop a fast and robust algorithm for the solution of high-order accurate discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured grids. Herein we present the development of a spectral hp-multigrid method, where the coarse ''grid'' levels are constructed by reducing the order (p) of approximation of the discretization using hierarchical basis functions (p-multigrid), together with the traditional (h-multigrid) approach of constructing coarser grids with fewer elements. On each level we employ variants of the element-Jacobi scheme, where the Jacobian entries associated with each element are treated implicitly (i.e., inverted directly) and all other entries are treated explicitly. The methodology is developed for the two-dimensional non-linear Euler equations on unstructured grids, using both non-linear (FAS) and linear (CGC) multigrid schemes. Results are presented for the channel flow over a bump and a uniform flow over a four element airfoil. Current results demonstrate convergence rates which are independent of both order of accuracy (p) of the discretization and level of mesh resolution (h).
International Journal for Numerical Methods in Fluids
Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈ 0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to a genuinely incompressible formulation. Our contributions to the state-of-the-art are twofold: Firstly, we present a high-performance discontinuous Galerkin solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures. The performance results presented in this work focus on the node-level performance and our results suggest that there is great potential for further performance improvements for current state-of-the-art discontinuous Galerkin implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the threedimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows.
Journal of Computational Physics, 2009
Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The algorithms are based on coupling both p-and h-multigrid (ph-multigrid) methods which are used in non-linear or linear forms, and either directly as solvers or as preconditioners to a Newton-Krylov method. The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton-GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem. It is concluded that the Newton-GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.
Journal of Computational Acoustics, 2013
A high-order interior penalty discontinuous Galerkin method for the compressible Navier-Stokes equations is introduced, which is a modification of the scheme given by Hartmann and Houston. In this paper we investigate the influence of penalization and boundary treatment on accuracy. By observing eigenvalues and condition numbers, a lower bound for the penalization term µ was found, whereas convergence studies depict reasonable upper bounds and a linear dependence on the critical time step size. By investigating conservation properties we demonstrate that different boundary treatments influence the accuracy by several orders of magnitude, and propose reasonable strategies to improve conservation properties.
High-order discontinuous Galerkin methods using an hp-multigrid approach
Journal of Computational Physics, 2006
The goal of this paper is to investigate and develop a fast and robust algorithm for the solution of high-order accurate discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured grids. Herein we present the development of a spectral hp-multigrid method, where the coarse ''grid'' levels are constructed by reducing the order (p) of approximation of the discretization using hierarchical basis functions (p-multigrid), together with the traditional (h-multigrid) approach of constructing coarser grids with fewer elements. On each level we employ variants of the element-Jacobi scheme, where the Jacobian entries associated with each element are treated implicitly (i.e., inverted directly) and all other entries are treated explicitly. The methodology is developed for the two-dimensional non-linear Euler equations on unstructured grids, using both non-linear (FAS) and linear (CGC) multigrid schemes. Results are presented for the channel flow over a bump and a uniform flow over a four element airfoil. Current results demonstrate convergence rates which are independent of both order of accuracy (p) of the discretization and level of mesh resolution (h).
Journal of Computational Physics, 2010
A reconstructed discontinuous Galerkin (RDG) method based on a Hierarchical WENO reconstruction, termed HWENO(P 1 P 2 ) in this work, designed not only to enhance the accuracy of discontinuous Galerkin method but also to ensure the nonlinear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this HWENO(P 1 P 2 ) method, a quadratic polynomial solution (P 2 ) is first reconstructed using a Hermite WENO (HWENO) reconstruction from the underlying linear polynomial (P 1 ) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact and consistent with the underlying DG method. The gradients (first moments) of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the RDG method. The developed HWENO(P 1 P 2 ) method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, robustness, and non-oscillatory property. The numerical experiments indicate that the HWENO(P 1 P 2 ) method is able to capture shock waves within one cell without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method.
The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier-Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior PenaltyGalerkin formulation. The non-linear terms in the Navier-Stokes equations are expressed in the convective form and approximated through the Lesaint-Raviart fluxes modified for DG methods. Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier-Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier-Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier-Stokes solver, we consider unsteady laminar flow pasta square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.
Development of a high order Discontinuous Galerkin Finite Element Solver
An unsteady high order Discontinuous Galerkin (DG) code has been developed, verified and validated for the solution of the two-dimensional incompressible Navier-Stokes equations. A second order stiffly stable method has been used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Interior Penalty formulation.Three variants, Symmetric Interior Penalty Galerkin (SIPG), Incomplete Interior Penalty Galerkin (IIPG) and Non Symmetric Interior Penalty Galerkin (NIPG), have been implemented and compared. The non-linear terms in the Navier-Stokes equations are expressed in the convective form and approximated through the Lesaint-Raviart fluxes modified for DG methods. The resulting method leads to a stable scheme for the unsteady Stokes and Navier-Stokes problems when equal order approximation is used for velocity and pressure and for all fluxes tested. For the full Navier-Stokes equations, two laminar test cases are considered for the square cylinder problem at Reynolds numbers of 10 (steady wake) and 100 (unsteady wake). The results are compared to the h/p Spectral code Nektar and the commercial Finite Volume code Fluent. The developed DG code shows similar convergence trends to Nektar for the test problems considered. For the unsteady wake case, the number of degrees of freedom necessary for Fluent to reach comparable accuracy is three times larger than for the two high order methods considered.