Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation

Barter, Garrett; Darmofal, David L.

doi:10.1016/j.jcp.2009.11.010

Cited by 182 publications

(148 citation statements)

References 49 publications

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“…The sub-cell resolution obtained by Persson and Peraire [102] using an artificial viscosity based approach is impressive. However, the extension of this approach by Barter and Darmofal [4,5] to avoid piecewise discontinuous artificial viscosity (which may generate spurious downstream errors) is somewhat involved. In summary, whilst a range of shock capturing approaches have been identified, there is further scope for identifying even cheaper, simpler and more robust schemes that offer sub-cell shock resolution, and which have a negligible impact on flow features away from shocks.…”

Section: Discussionmentioning

confidence: 99%

“…Finally, it should be noted that a recent extension of the technique developed by Persson and Peraire [102] has been suggested by Barter and Darmofal [4,5], who recognized that use of an element-wise constant artificial viscosity may induce oscillations in state gradients; which may pollute the downstream flow solution. To rectify this problem, Barter and Darmofal proposed a more complex formulation, which involves solving an additional partial differential equation to obtain a smooth artificial viscosity profile.…”

Section: Artificial Viscosity Based Approachesmentioning

confidence: 99%

“…Other than the method of Persson and Peraire [102], and its extension by Barter and Darmofal [4,5], the approaches detailed above (at least as applied) do not result in sub-cell resolution of shocks. This can be a significant problem if the mesh contains relatively large high-order elements.…”

Section: Automatic Mesh Adaptationmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

Section: Artificial Viscosity Based Approachesmentioning

confidence: 99%

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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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“…Per element the decision to refine is based on a smoothness indicator. Spurious oscillations in the neighbourhood of discontinuities are suppressed by adding artificial viscosity to the model [17,18] .…”

Section: Introductionmentioning

confidence: 99%

Efficient simulation of one-dimensional two-phase flow with a high-order h-adaptive space-time Discontinuous Galerkin method

et al. 2017

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a b s t r a c tOne-dimensional models for multiphase flow in pipelines are commonly discretised using first-order Finite Volume (FV) schemes, often combined with implicit time-integration methods. While robust, these methods introduce much numerical diffusion depending on the number of grid points. In this paper we propose a high-order, space-time Discontinuous Galerkin (DG) Finite Element method with h -adaptivity to improve the efficiency of one-dimensional multiphase flow simulations. For smooth initial boundary value problems we show that the DG method converges with the theoretical rate and that the growth rate and phase shift of small, harmonic perturbations exhibit superconvergence. We employ two techniques to accurately and efficiently represent discontinuities. Firstly artificial diffusion in the neighbourhood of a discontinuity suppresses spurious oscillations. Secondly local mesh refinement allows for a sharper representation of the discontinuity while keeping the amount of work required to obtain a solution relatively low. The proposed DG method is shown to be superior to FV.

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“…Recently, some authors have also recovered the artificial diffusion technique, obtaining subcell resolution (see, e.g. [13,14]). In [15], the same authors proposed an artificial diffusion of order O.h k / with 1 6 k 6 p, which for p high enough improves the threshold h=p.…”

mentioning

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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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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Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation

Cited by 182 publications

References 49 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

Efficient simulation of one-dimensional two-phase flow with a high-order h-adaptive space-time Discontinuous Galerkin method

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

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