p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Fidkowski, Krzysztof J.; Oliver, Todd; Lu, James; Darmofal, David L.

doi:10.1016/j.jcp.2005.01.005

Cited by 321 publications

(246 citation statements)

References 28 publications

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“…Implicit LU-SGS is also successfully used for SD and SV, such as the methods in [107,108]. P-multigrid approach, where p is the order of polynomial degree, is widely employed in DG [111,112], SV [108], SD [113], SE [114], and some similar schemes to accelerate convergence rate. It is worth to note that Kannan et al [107,108,113] have successfully blended p-multigrid approach with pre-conditions or implicit LU-SGS, and the convergence rate is drastically improved for bad and skewed unstructured grids.…”

Section: High-order and High Accurate Cfd Methods For Complex Grid Prmentioning

confidence: 99%

“…It is worth to note that Kannan et al [107,108,113] have successfully blended p-multigrid approach with pre-conditions or implicit LU-SGS, and the convergence rate is drastically improved for bad and skewed unstructured grids. For more details about p-multigrid methods, please refer to [108,[111][112][113][114] and the references therein.…”

Section: High-order and High Accurate Cfd Methods For Complex Grid Prmentioning

confidence: 99%

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High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

Deng

Mao

et al. 2012

Commun. comput. phys.

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Abstract. The purpose of this article is to summarize our recent progress in high-order and high accurate CFD methods for flow problems with complex grids as well as to discuss the engineering prospects in using these methods. Despite the rapid development of high-order algorithms in CFD, the applications of high-order and high accurate methods on complex configurations are still limited. One of the main reasons which hinder the widely applications of these methods is the complexity of grids. Many aspects which can be neglected for low-order schemes must be treated carefully for high-order ones when the configurations are complex. In order to implement highorder finite difference schemes on complex multi-block grids, the geometric conservation law and block-interface conditions are discussed. A conservative metric method is applied to calculate the grid derivatives, and a characteristic-based interface condition is employed to fulfil high-order multi-block computing. The fifth-order WCNS-E-5 proposed by Deng [9, 10] is applied to simulate flows with complex grids, including a double-delta wing, a transonic airplane configuration, and a hypersonic X-38 configuration. The results in this paper and the references show pleasant prospects in engineering-oriented applications of high-order schemes.

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Section: High-order and High Accurate Cfd Methods For Complex Grid Prmentioning

confidence: 99%

Section: High-order and High Accurate Cfd Methods For Complex Grid Prmentioning

confidence: 99%

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

Deng

Mao

et al. 2012

Commun. comput. phys.

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“…In 2005 Fidkowski et al [36] presented a p-multigrid algorithm in a DG context for solving the Navier-Stokes equations. The scheme utilized an element line Jacobi smoother.…”

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Vincent

Jameson

2011

Math. Model. Nat. Phenom.

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Abstract. Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of high-order methods amongst a non-specialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat towards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

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“…Furthermore LMA restricts the choice of preconditioners: for example an incomplete LU factorisation with fill-in cannot be applied, because the layout can not accommodate the additional nonzeros. However, a geometric or p-Multigrid scheme [10] could be a good choice in combination with a Jacobi smother; the LMA format has the geometric information readily available and an element structure can be maintained on the coarser levels with restriction schemes such as shown in Figure 28. The implementation of such a multigrid preconditioner is out of scope of this paper and will be addressed by future research.…”

Section: Preconditioningmentioning

confidence: 99%

Finite Element Algorithms and Data Structures on Graphical Processing Units

Reguly

Giles²

2013

Int J Parallel Prog

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The finite element method (FEM) is one of the most commonly used techniques for the solution of partial differential equations on unstructured meshes. This paper discusses both the assembly and the solution phases of the FEM with special attention to the balance of computation and data movement. We present a GPU assembly algorithm that scales to arbitrary degree polynomials used as basis functions, at the expense of redundant computations. We show how the storage of the stiffness matrix affects the performance of both the assembly and the solution. We investigate two approaches: global assembly into the CSR and ELLPACK matrix formats and matrix-free algorithms, and show the trade-off between the amount of indexing data and stiffness data. We discuss the performance of different approaches in light of the implicit caches on Fermi GPUs and show a speedup over a twosocket 12-core CPU of up to 10 times in the assembly and up to 6 times in the solution phase. We present our sparse matrix-vector multiplication algorithms that are part of a conjugate gradient iteration and show that a matrix-free approach may be up to two times faster than global assembly approaches and up to 4 times faster than NVIDIA's cuSPARSE library, depending on the preconditioner used.

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p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Cited by 321 publications

References 28 publications

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Finite Element Algorithms and Data Structures on Graphical Processing Units

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