A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids

Luo, Hong; Xia, Yidong; Spiegel, Seth C.; Nourgaliev, Robert; Jiang, Zonglin

doi:10.1016/j.jcp.2012.11.026

Cited by 98 publications

(49 citation statements)

References 39 publications

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“…More importantly, such numerical methods must be efficient on general unstructured meshes in multiple space dimensions. Several of such methods are available under the key names Discontinuous Galerkin (DG) methods [19,[21][22][23][24]40], ENO, WENO and HWENO methods (Weighted/Hermite Essentially Non Oscillatory) [1,4,9,16,50,52,55,57,58,[68][69][70]76,85,92], ADER schemes [2,8,9,[37][38][39]59,74,75,83,84,86,88,91], P N P M schemes [32,34,62,63], etc.…”

Section: Introduction and Contextmentioning

confidence: 99%

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Loubère¹,

Dumbser²,

Diot³

2014

Commun. comput. phys.

102

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International audienceIn this paper, we investigate the coupling of the Multi-dimensional Optimal Order De- tection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decre- menting processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic par- tial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements

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Section: Introduction and Contextmentioning

confidence: 99%

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Loubère¹,

Dumbser²,

Diot³

2014

Commun. comput. phys.

102

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“…Mathematical Problems in Engineering The results are not as good as those in the literature [39], because the shock detector is used, which directly affect the accuracy of solutions on smooth region. How to accurately judge problem units will be our future efforts.…”

Section: Transonic Flow Over Onera M6 Wingmentioning

confidence: 58%

Predictor-Corrector LU-SGS Discontinuous Galerkin Finite Element Method for Conservation Laws

Liu

Xie

2015

Mathematical Problems in Engineering

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Efficient implicit predictor-corrector LU-SGS discontinuous Galerkin (DG) approach for compressible Euler equations on unstructured grids is investigated by adding the error compensation of high-order term. The original LU-SGS and GMRES schemes for DG method are discussed. Van Albada limiter is employed to make the scheme monotone. The numerical experiments performed for the transonic inviscid flows around NACA0012 airfoil, RAE2822 airfoil, and ONERA M6 wing indicate that the present algorithm has the advantages of low storage requirements and high convergence acceleration. The computational efficiency is close to that of GMRES scheme, nearly 2.1 times greater than that of LU-SGS scheme on unstructured grids for 2D cases, and almost 5.5 times greater than that of RK4 on unstructured grids for 3D cases.

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“…First, cell-averaged variables and their derivatives are handily available in this formulation. This makes the implementation of reconstruction schemes and Hermite WENO limiters straightforward and simple [39,40]. Secondly, the Taylor basis is hierarchic.…”

Section: A Direct Dg Methods For Rans Equations With Sa Modelmentioning

confidence: 99%

A Direct Discontinuous Galerkin Method for Computation of Turbulent Flows on Hybrid Grids

Cheng

Liu

Yang

et al. 2016

46th AIAA Fluid Dynamics Conference

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A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible Reynolds-averaged Navier-Stokes (RANS) equations with a modified Spalart-Allmaras (SA) model on hybrid grids. The characteristic face length in the numerical flux functions defined by the DDG method is carefully examined and re-evaluated to take into consideration the fact that grids in near-wall regions for high Reynolds number flows are generally curved and highly stretched. The effect with respect to the different choices of penalty parameter in simulating turbulent flows is also discussed in this work. A number of typical test cases are presented to demonstrate the performance of applying DDG method for high Reynolds number turbulent flows. Numerical results obtained demonstrate that the developed DDG method is able to deliver satisfactory and reliable results for simulating high Reynolds number turbulent flows and indicates its potential for practical engineering applications.

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A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids

Cited by 98 publications

References 39 publications

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Predictor-Corrector LU-SGS Discontinuous Galerkin Finite Element Method for Conservation Laws

A Direct Discontinuous Galerkin Method for Computation of Turbulent Flows on Hybrid Grids

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