A Comparison of Higher-Order Methods on a Set of Canonical Aerodynamics Applications (original) (raw)
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International Journal for Numerical Methods in Engineering, 2009
This paper presents a comparison between two high order methods. The first one is a high-order Finite Volume (FV) discretization on unstructured grids that uses a meshfree method (Moving Least Squares (MLS)) in order to construct a piecewise polynomial reconstruction and evaluate the viscous fluxes. The second method is a discontinuous Galerkin (DG) scheme. Numerical examples of inviscid and viscous flows are presented and the solutions are compared. The accuracy of both methods, for the same grid resolution, is similar, although the finite volume scheme is consistently more accurate in the present tests. Furthermore, the DG scheme requires a larger number of degrees of freedom than the FV-MLS method.
AIAA Paper, 2006
The development of optimal, or near optimal solution strategies for higher-order discretizations, including steady-state solutions methodologies, and implicit time integration strategies, remains one of the key determining factors in devising higher-order methods which are not just competitive but superior to lower-order methods in overall accuracy and efficiency. The goal of this work 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 extend our previous work to the three-dimensional steady-state Euler equations, by coupling the spectral p-multigrid approach with a more traditional agglomeration h-multigrid method for hybrid meshes, in a full-multigrid iteration strategy. In this hp-multigrid approach 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. The overall goal is the development of a solution algorithm which delivers convergence rates which are independent of "p" (the order of accuracy of the discretization) and independent of "h" (the degree of mesh resolution), while minimizing the cost of each iteration. The investigation of efficient smoothers to be used at each level of the multigrid algorithm is also pursued, and comparisons between different integration strategies are made as well. Current three-dimensional results demonstrate convergence rates which are independent of both order of accuracy (p) of the discretization and level of mesh resolution (h).
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
The goal of this paper is to investigate and develop fast and robust solution techniques for high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured meshes. Previous work was focused on the development of hp-multigrid techniques for inviscid flows and the current work concentrates on the extension of these solvers to steady-state viscous flows including the effects of highly anisotropic hybrid meshes. Efficiency and robustness are improved through the use of mixed triangular and quadrilateral mesh elements, the formulation of local order-reduction techniques, the development of a line-implicit Jacobi smoother, and the implementation of a Newton-GMRES solution technique. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear multigrid schemes. Results are presented for a flat plate boundary layer and for flow over a NACA0012 airfoil and a two-element airfoil. Current results demonstrate convergence rates which are independent of the degree of mesh anisotropy, order of accuracy (p) of the discretization and level of mesh resolution (h). Additionally, preliminary results of on-going work for the extension to the Reynolds Averaged Navier-Stokes(RANS) equations and the extension to three dimensions are given. agglomeration multigrid technique (h-multigrid) for Discontinuous Galerkin discretizations. This paper represents an extension to the Navier-Stokes equations of the previous work originally presented in references. The extension includes the DG discretization of the viscous fluxes using the Symmetric Interior Penalty Method (SIP) together with the capability of using hybrid meshes on two-dimensional configurations. In order to maintain the convergence rate of the hp-multigrid solver on hybrid anisotropic meshes, a line creation algorithm and line-implicit Jacobi smoother have been devised and implemented . Additional solver efficiency is obtained through the implementation of a preconditioned Newton-GMRES Krylov method, which represents an alternative method for solving the linear system given by Newton's method. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear hp-multigrid schemes as preconditioners. While results presented here pertain only to steady-state problems, the methods are easily extended to time-dependent problems. 5 Extensions to the Reynolds-averaged Navier-Stokes equations, involving the solution of an additional turbulence modeling equation are investigated.
Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
With high-order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. One of the most recent methods to be developed is the flux reconstruction approach, which allows various well-known high-order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the more widely adopted discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretization. By considering both linear and nonlinear advection equations on a regular grid, we examine the mathematical properties that connect these discretizations. These arguments are further confirmed by the results of an empirical numerical study.
Journal of Computational Physics, 2011
We develop a new hierarchical reconstruction (HR) method for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang . The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes. ‡
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).
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).
A Detailed Comparison of Weno and SFV High-Order Methods for Inviscid Aerodynamic Applications
2014
The purpose of this work is to compare two numerical formulations for unstructured grids that achieve high-order spatial discretization for compressible aerodynamic flows. High-order methods are necessary on the analysis of complex flows to reduce the number of mesh elements one would otherwise need if using traditional second-order schemes. In the present work, the 2-D Euler equations are solved numerically in a finite volume, cell centered context. The third-order Weighted Essentially Non-Oscillatory (WENO) and Spectral Finite Volume (SFV) methods are considered in this study for the spatial discretization of the governing equations. Time integration uses explicit, Runge-Kutta type schemes. Two literature test cases, one steady and one unsteady, are considered to assess the resolution capabilities and performance of the two spatial discretization methods. Both methods are suitable for the aerospace applications of interest. However, each method has characteristics that excel over the other scheme. The present results are valuable as a form of providing guidelines for future developments regarding high-order methods for unstructured meshes.
A class of Reconstructed Discontinuous Galerkin Methods for Compressible Flows on Arbitrary Grids
49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2011
A class of reconstructed discontinuous Galerkin (DG) methods is presented to solve compressible flow problems on arbitrary grids. The idea is to combine the efficiency of the reconstruction methods in finite volume methods and the accuracy of the DG methods to obtain a better numerical algorithm in computational fluid dynamics. The beauty of the resulting reconstructed discontinuous Galerkin (RDG) methods is that they provide a unified formulation for both finite volume and DG methods, and contain both classical finite volume and standard DG methods as two special cases of the RDG methods, and thus allow for a direct efficiency comparison. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are presented to obtain a quadratic polynomial representation of the underlying linear discontinuous Galerkin solution on each cell via a so-called in-cell reconstruction process. The devised in-cell reconstruction is aimed to augment the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution. These three reconstructed discontinuous Galerkin methods are used to compute a variety of compressible flow problems on arbitrary meshes to assess their accuracy. The numerical experiments demonstrate that all three reconstructed discontinuous Galerkin methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstructed DG method provides the best performance in terms of both accuracy, efficiency, and robustness.
An unstructured grid implementation of high-order spectral finite volume schemes
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2010
The present work implements the spectral finite volume scheme in a cell centered finite volume context for unstructured meshes. The 2-D Euler equations are considered to represent the flows of interest. The spatial discretization scheme is developed to achieve high resolution and computational efficiency for flow problems governed by hyperbolic conservation laws, including flow discontinuities. Such discontinuities are mainly shock waves in the aerodynamic studies of interest in the present paper. The entire reconstruction process is described in detail for the 2 nd to 4 th order schemes. Roe's flux difference splitting method is used as the numerical Riemann solver. Several applications are performed in order to assess the method capability compared to data available in the literature. The results obtained with the present method are also compared to those of essentially nonoscillatory and weighted essentially non-oscillatory high-order schemes. There is a good agreement with the comparison data and efficiency improvements have been observed.