Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes (original) (raw)
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2016
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. The flux reconstruction approach allows various well-known high order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the 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 t...
This paper investigates the connections between many popular variants of the well-established discontinuous Galerkin method and the recently developed high-order flux reconstruction approach on irregular tensor-product grids. We explore these connections by analysing three nodal versions of tensor-product discontinuous Galerkin spectral element approximations and three types of flux reconstruction schemes for solving systems of conservation laws on irregular tensor-productmeshes.We demonstrate that the existing connections established on regular grids are also valid on deformed and curved meshes for both linear and nonlinear problems, provided that the metric terms are accounted for appropriately. We also find that the aliasing issues arising from nonlinearities either due to a deformed/curved elements or due to the nonlinearity of the equations are equivalent and can be addressed using the same strategies both in the discontinuous Galerkin method and in the flux reconstruction approach. In particular, we show that the discontinuous Galerkin and the flux reconstruction approach are equivalent also when using higher-order quadrature rules that are commonly employed in the context of over- or consistent-integration-based dealiasing methods.The connections found in this work help to complete the picture regarding the relations between these two numerical approaches and show the possibility of using over- or consistent-integration in an equivalent manner for both the approaches.
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).
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.
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).
2010
The direct discontinuous Galerkin (DDG) method was developed by Liu and Yan to discretize the diffusion flux. It was implemented for the discontinuous Galerkin (DG) formulation. In this paper, we perform four tasks: (i) implement the direct discontinuous Galerkin (DDG) scheme for the spectral volume method (SV) method, (ii) design and implement two variants of DDG (called DDG2 and DDG3) for the SV method, (iii) perform a Fourier type analysis on both methods when solving the 1D diffusion equation and combine the above with a non-linear global optimizer, to obtain modified constants that give significantly smaller errors (in 1D), (iv) use the above coefficients as starting points in 2D. The dissipation properties of the above schemes were then compared with existing flux formulations (local discontinuous Galerkin, Penalty and BR2). The DDG, DDG2 and DDG3 formulations were found to be much more accurate than the above three existing flux formulations. The accuracy of the DDG scheme is heavily dependent on the penalizing coefficient for the odd ordered schemes. Hence a loss of accuracy was observed even for mildly non-uniform grids for odd ordered schemes. On the other hand, the DDG2 and DDG3 schemes were mildly dependent on the penalizing coefficient for both odd and even orders and retain their accuracy even on highly irregular grids. Temporal analysis was also performed and this yielded some interesting results. The DDG and its variants were implemented in 2D (on triangular meshes) for Navier-Stokes equations. Even the non-optimized versions of the DDG displayed lower errors than the existing schemes (in 2D). In general, the DDG and its variants show promising properties and it indicates that these approaches have a great potential for higher dimension flow problems.
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).
A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals
International Journal for Numerical Methods in Fluids, 2010
This work presents a study on the performance of nodal bases on triangles and on quadrilaterals for discontinuous Galerkin solutions of hyperbolic conservation laws. A nodal basis on triangles and two tensor product nodal bases on quadrilaterals are considered. The quadrilateral element bases are constructed from the Lagrange interpolating polynomials associated with the Legendre-Gauss-Lobatto points and from those associated with the classical Legendre-Gauss points. Settings of interest concern the situation in which a mesh of triangular elements is obtained by dividing each quadrilateral element into two triangular elements or vice versa, the mesh of quadrilateral elements is obtained by merging two adjacent triangular elements. To assess performance, we use a linear advecting rotating plume transport problem as a test case. For cases where the order of the basis is low to moderate, the computing time used to reach a given final time for the quadrilateral elements is shorter than that for the triangular elements. The numerical results also show that the quadrilateral elements yield higher computational efficiency in terms of cost to achieve similar accuracy.
A comparative study of different reconstruction schemes for a reconstruction-based discontinuous Galerkin, termed RDG(P1P2) method is performed for compressible flow problems on arbitrary grids. The RDG method is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution via a reconstruction scheme commonly used in the finite volume method. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are implemented to obtain a quadratic polynomial representation of the underlying discontinuous Galerkin linear polynomial solution on each cell. These three reconstruction/recovery methods are compared for a variety of compressible flow problems on arbitrary meshes to access their accuracy and robustness. The numerical results demonstrate that all three reconstruction methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy and robustness.
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.