A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids

Wang, Zhi Jian; Gao, Haiyang

doi:10.1016/j.jcp.2009.07.036

Cited by 405 publications

(196 citation statements)

References 51 publications

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“…Two approaches are developed in [45]. One is the Lagrange polynomial (LP) approach which approximate the (nonlinear) flux vecter with degree interpolation polynomials…”

Section: Review Of the Correction Procedures Via Reconstruction Formulmentioning

confidence: 99%

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A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

Haga

Gao

Wang

2011

Math. Model. Nat. Phenom.

127

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Abstract. The newly developed unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be completely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compressible Navier-Stokes equations to demonstrate its performance.

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“…Two approaches are developed in [45]. One is the Lagrange polynomial (LP) approach which approximate the (nonlinear) flux vecter with degree interpolation polynomials…”

Section: Review Of the Correction Procedures Via Reconstruction Formulmentioning

confidence: 99%

“…Recently, a novel formulation named correction procedure via reconstruction (CPR) was developed by Huynh [16,17] for 1D conservation laws, and extended to simplex and hybrid meshes by Wang and Gao [45]. The CPR method is based on a nodal differential form, with an elementwise discontinuous polynomial solution space.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

Haga

Gao

Wang

2011

Math. Model. Nat. Phenom.

127

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“…Using the weak formulation precludes the need to represent the flux in a polynomial space when evaluating volume contributions to the residual; the flux is computed directly at the cubature nodes as required in the volume integral evaluation. This reduces aliasing errors as compared to discretizations based on the doubly integrated by parts weak form [3] in which aliasing errors are typically addressed through the use of the chain rule approach [6].…”

Section: Code and Discretization Descriptionsmentioning

confidence: 99%

On the Necessity of Superparametric Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries

Zwanenburg

Nadarajah

2017

23rd AIAA Computational Fluid Dynamics Conference

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We provide numerical evidence demonstrating the necessity of employing a superparametric geometry representation in order to obtain optimal convergence orders on two-dimensional domains with curved boundaries when solving the Euler equations using Discontinuous Galerkin methods. However, concerning the obtention of optimal convergence orders for the Navier-Stokes equations, we demonstrate numerically that the use of isoparametric geometry representation is sufficient for the case considered here.

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“…Recently, Huynn proposed the flux reconstruction (FR) method, 6,7 which provides an unifying framework of the above-mentioned methods (DG, SV and SD). Furthermore, Wang introduced a lifting collocation operator 8 to deal with multi-dimensional problems on simplex and mixed grids. The two approaches, combined and renamed as correction procedures via reconstruction (CPR), share many merits of higher-order methods in terms of accuracy and stability, while it is more simple and efficient.…”

Section: -3mentioning

confidence: 99%

Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Park

Chang

Kim

2014

52nd Aerospace Sciences Meeting

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Higher-order multi-dimensional limiting Process (MLP) [J. S. Park and C. Kim, Higherorder Multi-dimensional Limiting Strategy for Discontinuous Galerkin Methods in Compressible Inviscid and Viscous Flows, Comp. & Fluids, In press] is improved and applied to correction procedure via reconstruction (CPR) on unstructured grids. MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn reconstruction. The resulting algorithms, called the hierarchical MLP, facilitate the accurate capturing of detailed flow structures in both continuous and discontinuous regions. This algorithm has been developed in the modal DG framework, but it also can be formulated into a nodal framework, most notably the CPR framework. Troubled-cells are detected by applying the MLP concept, and the final accuracy is determined by the projection procedure and MLP limiting step. Through extensive numerical analyses and computations, it is demonstrated that the proposed limiting approach yields the desired accuracy and outstanding performances in resolving compressible inviscid and viscous flow features.

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A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids

Cited by 405 publications

References 51 publications

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

On the Necessity of Superparametric Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries

Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

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