Runge–Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes

Zhu, Jun; Qiu, Jianxian; Shu, Chi‐Wang; Dumbser, Michael

doi:10.1016/j.jcp.2007.12.024

Cited by 183 publications

(91 citation statements)

References 29 publications

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“…The shocks and the rarefaction fan, which is created around the corner, are well-captured without generating relevant noise. This is specially remarkable around the upper slip line from the triple point and behind the Mach stem, compared to other results [5,20]. The contact discontinuity arising from the triple point is slightly better resolved with the high-order approximation despite fewer DOFs are involved.…”

Section: The Forward-facing Step Problemmentioning

confidence: 53%

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A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

Huerta

Casoni

Peraire

2011

Numerical Methods in Fluids

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This is the accepted version of the following article: [Huerta, A., Casoni, E. and Peraire, J. (2012), A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int. J. Numer. Meth. Fluids, 69: 1614–1632. doi:10.1002/fld.2654], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.2654/fullThis article presents a novel shock-capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high-order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high-order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach.Peer ReviewedPostprint (author's final draft

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Section: The Forward-facing Step Problemmentioning

confidence: 53%

“…It is now a standard numerical benchmark (see, e.g. [5,20,[45][46][47] An inflow boundary condition is applied at the left end of the computational domain and outflow boundary condition at the right end. Along the walls of the tunnel, and on the boundary marked by the step, inviscid wall boundary conditions are applied.…”

Section: The Forward-facing Step Problemmentioning

confidence: 99%

A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

Huerta

Casoni

Peraire

2011

Numerical Methods in Fluids

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“…A full review of all such combinations is beyond the scope of this article. However, a notable example is the application of WENO type limiters in a DG context by Qiu and Shu [109] and Zhu et al [148].…”

Section: Limiter Based Approachesmentioning

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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“…One promising direction of research is design of DG limiters based on weighted essentially non-oscillatory (WENO) finite volume methodology. The approach was developed in [42] on structured meshes and was further extended to unstructured meshes in [32,54].…”

Section: Nonphysical Oscillations In Higher Order Dg Schemesmentioning

confidence: 99%

Application of a Higher Order Discontinuous Galerkin

Wolkov

Hirsch

Petrovskaya

2011

Math. Model. Nat. Phenom.

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Abstract. We discuss the issues of implementation of a higher order discontinuous Galerkin (DG) scheme for aerodynamics computations. In recent years a DG method has intensively been studied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has been designed for numerical solution of the 3-D Euler and Navier-Stokes equations. Our discussion is mainly based on the results of the DG study conducted in TsAGI in collaboration with the NUMECA International. The capacity of a DG scheme to tackle challenging computational problems is demonstrated and its potential advantages over FV schemes widely used in modern computational aerodynamics are highlighted.

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Runge–Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes

Cited by 183 publications

References 29 publications

A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

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

Application of a Higher Order Discontinuous Galerkin

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