Dealiasing techniques for high-order spectral element methods on regular and irregular grids (original) (raw)
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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).
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.
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.
High-Order Finite-Volume Methods on Locally-Structured Grids
2009
We describe recent developments in high-order (greater than second-order) accurate finite-volume discretizations of partial differential equations (PDE) in divergence form. We will address a number of algorithmic issues that arise in these methods, including the choice of limiters for hyperbolic problems that preserve high-order accuracy; defining quadrature rules for fluxes on mapped grids that are freestream-preserving; and high-order interpolation of ghost cell values at block or refinement boundaries.
Journal of Computational Physics, 2021
This paper presents eigensolution and non-modal analyses for immersed boundary methods (IBMs) based on volume penalization for the linear advection equation. This approach is used to analyze the behavior of flux reconstruction (FR) discretization, including the influence of polynomial order and penalization parameter on numerical errors and stability. Through a semi-discrete analysis, we find that the inclusion of IBM adds additional dissipation without changing significantly the dispersion of the original numerical discretization. This agrees with the physical intuition that in this type of approach, the solid wall is modelled as a porous medium with vanishing viscosity. From a stability point view, the selection of penalty parameter can be analyzed based on a fully-discrete analysis, which leads to practical guidelines on the selection of penalization parameter. Numerical experiments indicate that the penalization term needs to be increased to damp oscillations inside the solid (i.e. porous region), which leads to undesirable time step restrictions. As an alternative, we propose to include a second-order term in the solid for the no-slip wall boundary condition. Results show that by adding a second-order term we improve the overall accuracy with relaxed time step restriction. This indicates that the optimal value of the penalization parameter and the second-order damping can be carefully chosen to obtain a more accurate scheme. Finally, the approximated relationship between these two parameters is obtained and used as a guideline to select the optimum penalty terms in a Navier-Stokes solver, to simulate flow past a cylinder.
A Comparison of Higher-Order Methods on a Set of Canonical Aerodynamics Applications
20th AIAA Computational Fluid Dynamics Conference, 2011
Higher-order discretizations have the potential to reduce the computational cost required to achieve a desired error level. In this study, we consider higher-order discretizations of the conservation equations suitable for unstructured, triangular grids. In particular, the methods studied include continuous (SUPG/GLS) and classical discontinuous Galerkin (DG) finite element methods, the correction procedure via reconstruction (CPR) formulations of the DG and spectral volume methods, and cell and vertex-centered finite volume (FV) algorithms. This paper presents subsonic and supersonic, inviscid results for a canonical set of aerodynamic applications. Error convergence and computational performance of these discretizations are compared, and preliminary results indicate that the methods perform relatively similarly. When singularities are present in the flow solutions and uniformly refined meshes are used, all methods fail to achieve optimal convergence rates, and the performance benefits of the higher-order discretizations are reduced; adaptive meshing improves the efficiency of the higher-order method and recovers optimal convergence rates.
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.
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...