A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids

Luo, Hong; Baum, Joseph D.; Löhner, Rainald

doi:10.1016/j.jcp.2005.06.019

Cited by 181 publications

(142 citation statements)

References 25 publications

(41 reference statements)

Supporting

Mentioning

141

Contrasting

Order By: Relevance

“…However much further work is needed before their efficiency is on a par with that of the best geometric multigrid approaches. Studies to date indicate that using a judicious choice of explicit and implicit smoothers at various multigrid levels may help to circumvent memory related issues whilst maintaining a suitable rate of convergence [85] [80]. Also, studies indicate that a combination of both p-multigrid and geometric multigrid may be beneficial [91].…”

Section: Discussionmentioning

confidence: 99%

“…Convergence rates independent of both order and mesh size were achieved. Also in 2006, Luo, Baum and Lohner [85] developed a p-multigrid scheme for compressible inviscid flow on unstructured grids. They investigated using a mix of explicit RK smoothers (for the highest-order multigrid levels) and matrix free implicit SGS and LU-SGS smoothers (for low-order multigrid levels), such that the scheme converged quickly, yet had reasonable memory requirements.…”

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

“…Finally in 2009, Liang, Kannan and Wang [80] presented a p-multigrid algorithm in a SD context to solve the Euler equations. The scheme utilized a mix of implicit and explicit smoothers at different multigrid levels (similar to the approach of Luo, Baum and Lohner [85]). Specifically, explicit smoothers were used at the highest-order levels to limit memory requirements, and implicit smoothers were used for the low-order levels to expedite convergence.…”

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

See 2 more Smart Citations

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

Vincent

Jameson

2011

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

Section: Geometric Multigrid and P-multigrid Methodsmentioning

confidence: 99%

See 1 more Smart Citation

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

Vincent

Jameson

2011

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

show abstract

“…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%

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

Deng

Mao

et al. 2012

Commun. comput. phys.

View full text Add to dashboard Cite

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.

show abstract

Implicit time integration of hyperbolic conservation laws via discontinuous Galerkin methods

Xin

Flaherty

2011

Int. J. Numer. Meth. Biomed. Engng.

View full text Add to dashboard Cite

SUMMARYA high-order, matrix-free implicit method has been developed for the transient solutions of hyperbolic conservation laws. The discontinuous Galerkin method is applied for temporal discretization. This method has the advantage that its discretization error is O( t 2 p+1 ) when a polynomial basis of degree p is used for time discretization. The nonlinear system of equations from the implicit time discretization is solved at each time step using a nonlinear Krylov subspace projection method. The system of linear equations is solved by the generalized minimum residual algorithm with a lower-upper symmetric Gauss-Seidel preconditioner. The numerical results from the inviscid Burgers' equation indicate that the implicit method is several times faster in performance relative to explicit integration by the total variation diminishing Runge-Kutta method of order 3. The forward Euler method requires a time step proportional to the square of the spatial step for stability with equations such as Burgers' equation (J. Sci. Comput. 2001; 16:173-261). It would, hence, be much less efficient than other explicit methods, for example, Cockburn and Shu (Math. Comput. 1989; 52:411-435), which would only require a time step proportional to the spatial step.

show abstract

A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids

Cited by 181 publications

References 25 publications

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

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

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

Implicit time integration of hyperbolic conservation laws via discontinuous Galerkin methods

Contact Info

Product

Resources

About