Discontinuous Galerkin Methods for Computational Fluid Dynamics

Cockburn, Bernardo

doi:10.1002/9781119176817.ecm2053

Cited by 16 publications

(13 citation statements)

References 284 publications

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“…There were those ones focused on the discretization of hyperbolic problems, using stable numerical fluxes over element interfaces, in the spirit of finite volume methods, as reviewed in the work of Cockburn. 42 A variety of other schemes were introduced for elliptic and parabolic problems as well, using interior penalty terms to weakly impose continuity in internal interfaces, in the fashion used in Nitsche's method for boundary conditions enforcement. The introduction of these stabilization mechanisms is helpful to damp spurious oscillations and to ensure optimal orders of convergence.…”

Section: Symmetric Interior Penalty Formulationmentioning

confidence: 99%

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

Devloo

Faria

Farias

et al. 2018

Numerical Meth Engineering

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Summary Finite element formulations for second‐order elliptic problems, including the classic H1‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L2‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.

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Section: Symmetric Interior Penalty Formulationmentioning

confidence: 99%

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

Devloo

Faria

Farias

et al. 2018

Numerical Meth Engineering

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“…Spatially high-order accurate discretizations demonstrate much smaller numerical errors than low-order methods, in terms of dispersion as well as dissipation [1][2][3][4]. Consequently, they tend to be more accurate per degree of freedom (DOF) [5,6].…”

Section: Introductionmentioning

confidence: 99%

Optimized Explicit Runge-Kutta Schemes for Entropy Stable Discontinuous Collocated Methods Applied to the Euler and Navier–Stokes equations

Jahdali

Boukharfane²,

Dalcín³

et al. 2021

AIAA Scitech 2021 Forum

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In this work, we design a new set of optimized explicit Runge-Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with entropy stable collocated discontinuous Galerkin methods. The optimization of the new time integration schemes is based on the spectrum of the discrete spatial operator for the advection equation. To demonstrate the efficiency and accuracy of the new schemes compared to some widely used classic explicit Runge-Kutta methods, we report the wall-clock time versus the error for the simulation of the two-dimensional advection equation and the propagation of an isentropic vortex with the compressible Euler equations. The efficiency and robustness of the proposed optimized schemes for more complex flow problems are presented for the three-dimensional Taylor-Green vortex at a Reynolds number of Re = 1.6 × 10 3 and Mach number Ma = 0.1, and the flow past two identical spheres in tandem at a Reynolds number of Re = 3.9 × 10 3 and Mach number Ma = 0.1.

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“…HDG inherits all the advantages of high-order Discontinous Galerkin (DG) methods [3,16,19,22] that have made them so popular in CFD in the last decade, such as local conservation of quantities of interest, intrinsic stabilization thanks to a proper definition of numerical fluxes at element boundaries, suitability for code vec-torization and parallel computation, and suitability for adaptivity. But, HDG outperforms other DG methods for problems involving self-adjoint operators, due to two main peculiar-ities: hybridization and superconvergence properties.…”

Section: Introductionmentioning

confidence: 99%

eXtended Hybridizable Discontinous Galerkin (X-HDG) for Void Problems

et al. 2015

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A strategy for the Hybridizable Discontinous Galerkin (HDG) solution of problems with voids, inclusions or free surfaces is proposed. It is based on an eXtended Finite Element philosophy with a level-set description of interfaces. Thus, the computational mesh is not required to fit the interface (i.e. the boundary), simplifying and reducing the cost of mesh generation and, in particular, avoiding continuous remeshing for evolving interfaces. Differently to previous proposals for HDG solution with non-fitting meshes, here the computational mesh covers the domain, avoiding extrapolations, and ensuring the robustness of the method. The local problem at elements not cut by the interface, and the global problem, are discretized as usual in HDG. A modified local problem is considered at elements cut by the interface. At every cut element, an auxiliary trace variable on the boundary is introduced, which is eliminated afterwards using the boundary conditions on the interface, keeping the original unknowns and the structure of the local problem solver. An efficient and robust methodology for numerical integration in cut elements, in the context of high-order approximations, is also proposed. Numerical experiments demonstrate how X-HDG keeps the optimal convergence, superconvergence, and accuracy of HDG with no need of adapting the computational mesh to the interface boundary.

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Discontinuous Galerkin Methods for Computational Fluid Dynamics

Cited by 16 publications

References 284 publications

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

Optimized Explicit Runge-Kutta Schemes for Entropy Stable Discontinuous Collocated Methods Applied to the Euler and Navier–Stokes equations

eXtended Hybridizable Discontinous Galerkin (X-HDG) for Void Problems

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