On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods

Yu, Meilin; Wang, Zhi Jian; Liu, Yen

doi:10.1016/j.jcp.2013.11.023

Cited by 103 publications

(54 citation statements)

References 39 publications

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“…Great efforts have been paid on the development of compact schemes in the past decades [21]. Two main representatives of compact schemes are the Discontinuous Galerkin (DG) method [3] and correction procedure via reconstruction (CPR) [29], with very restricted CFL number in the determination of time step. Most of those methods use Riemann solvers or approximate Riemann solvers for the flux evaluation, and adopt the Runge-Kutta time stepping for the time accuracy.…”

Section: Introductionmentioning

confidence: 99%

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

Zhao

Shyy

et al. 2020

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

Zhao

Shyy

et al. 2020

Journal of Computational Physics

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“…The latter method combines high order accuracy with the ability to handle complex geometries described by hybrid unstructured meshes by incorporating a minimal computational stencil. However, the computational efficiency of this method (alongside the spectral volume [8,10] and spectral difference [8,11,12] methods is generally believed to be inferior to more commonly used methods as the Finite Differences (FD) and the Finite Volume (FV) [13,14] methods. Solution adaptive refinement strategies of h/p-type can reduce the computational time for high resolution simulations of complex flows, using various methods, including the DG method, without compromising numerical accuracy [15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

A new approach to modelling the two way coupling for momentum transfer in a hollow-cone spray

Papoutsakis

Sazhin

Begg

et al. 2017

Proceedings ILASS–Europe 2017. 28th Conference on Liquid Atomization and Spray Systems

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A new approach to modelling the interaction between droplets and the carrier phase is suggested. The new model is applied to the analysis of a spray injected into a chamber of quiescent air, using an Eulerian-Lagrangian approach. The conservative formulation of the equations for mass, momentum and energy transport is used for the analysis of the carrier phase. The dispersed phase is modelled using the Lagrangian approach with droplets represented by individual parcels. The implementation of the Discontinuous Galerkin method (ForestDG), based on a topological representation of the computational mesh by a hierarchical structure consisting of oct-quad-and binary trees, is used in our analysis. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution for the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of the accuracy of discretisation of shear layers and vortices. The initial domain consists of a graph of unitarian-trees representing hexahedral, prismatic and tetrahedral elements. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. These are attached to the tree data structure which facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree. This enables us to refine the computational mesh in the vicinity of the droplet parcels aiming to accurately resolve the coupling between the two phases. KeywordsDroplets, Sprays, Vortex Rings, Discontinuous Galerkin, Adaptive Mesh Refinement. IntroductionThe need to accurately model the interaction between droplets and carrier phase (coupling) in various engineering applications is well known [1]. Although various approaches to modelling this coupling have been suggested [1,2,3,4,5], this problem is far from being solved. The main focus of this paper is on the description of the new approach to this coupling and the application of this approach to modelling realistic sprays in gasoline-engine-like conditions. This new approach is based on the application of the adaptive mesh refinement in the vicinity of the droplet parcels. The Discontinuous Galerkin (DG) method [6,7,8,9] is used for solving the equations for the carrier phase. The latter method combines high order accuracy with the ability to handle complex geometries described by hybrid unstructured meshes by incorporating a minimal computational stencil. However, the computational efficiency of this method (alongside the spectral volume [8,10] and spectral difference [8,11,12] methods is generally believed to be inferior to more commonly used methods as the Finite Differenc...

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“…The discontinuous Galerkin (DG), 2-7 the spectral volume (SV), 4,8 and the spectral difference 4,9,10 methods have shown promise for high resolution computations of complex flows because they have a compact stencil, and retain the design order of accuracy even for meshes of moderate quality that would often result from grid generation over complex three dimensional configurations. The potential applications of these methods for high resolution simulations of practical flow problems can further be enhanced with the use of mixed-type meshes and solution adaptive schemes.…”

Section: Introductionmentioning

confidence: 99%

“…There are several implementations of the DG methods depending on how the degrees-of-freedom are chosen. 10 The nodal DG implementation with Lagrange polynomial basis functions 31, 32 and the more effective extension of Hesthaven and Worburton 33,34 are the most efficient. The filtering approach was tested with nodal bases 35 however it was found that the extension to unstructured meshes was not a trivial task.…”

Section: Introductionmentioning

confidence: 99%

A dissipative Filter for the Discontinuous Galerkin method

Panourgias

Ekaterinaris²

2015

53rd AIAA Aerospace Sciences Meeting

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Complex three dimensional flows in domains with nontrivial geometry require use of mixed type unstructured meshes. The discontinuous Galerkin (DG) method is an unstructured grid method and development of a unified approach for discontinuity capturing with the DG method when higher order expansions are employed is therefore required. A systematic procedure for incorporating a dissipative mechanism to DG discretizations in unstructured meshes is developed. This adaptive non-linear dissipative mechanism enables capturing discontinuities and steep flow gradients without numerical oscillations. A filter designed and extensively used in the development of low dissipative well-balanced high order accurate finite difference schemes is employed. The filtering operator is adapted to the finite element context of DG discretizations. The filter operator is constructed and applied in the canonical computational domain where it can be applied in a direction per direction basis and then extended for all element types in unstructured meshes using collapsed coordinate transformations. The performance of the proposed dissipative mechanism is demonstrated and evaluated for different orders of expansions for one-dimensional and multidimensional problems with exact solutions. It is further applied for a number of standard inviscid flow test problems including strong shocks interactions to demonstrate the potential of the method. The dissipative mechanism yields results comparable to three-dimensional TVB and hierarchical limiting. Furthermore, the proposed approach is suitable for h/p-adaptivity in order to locally enhance resolution for three dimensional flow simulations that include discontinuities and complex flow features.

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On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods

Cited by 103 publications

References 39 publications

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

A new approach to modelling the two way coupling for momentum transfer in a hollow-cone spray

A dissipative Filter for the Discontinuous Galerkin method

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