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A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and NavierStokes Equations
"... In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and NavierStokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuit ..."
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Cited by 49 (13 self)
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In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and NavierStokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, but offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method. I.
A Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems,” submitted
 SIAM J. for Numerical Analaysis
, 2006
"... Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is the ..."
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Cited by 34 (14 self)
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Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility.
Facilitating the adoption of unstructured highorder methods amongst a wider community of fluid dynamicists
 Mathematical Modelling of Natural Phenomena
"... Abstract. Theoretical studies and numerical experiments suggest that unstructured highorder methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing highorder schemes are generally less robust and more complex ..."
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Cited by 12 (5 self)
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Abstract. Theoretical studies and numerical experiments suggest that unstructured highorder methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing highorder schemes are generally less robust and more complex to implement than their loworder counterparts. These issues, in conjunction with difficulties generating highorder meshes, have limited the adoption of highorder techniques in both academia (where the use of loworder schemes remains widespread) and industry (where the use of loworder schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of highorder methods amongst a nonspecialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured highorder meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of highorder 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 highorder schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework.
Scalable Parallel NewtonKrylov Solvers for Discontinuous Galerkin discretizations.” AIAA
, 2009
"... We present techniques for implicit solution of discontinuous Galerkin discretizations of the NavierStokes equations on parallel computers. While a blockJacobi method is simple and straightforward to parallelize, its convergence properties are poor except for simple problems. Therefore, we conside ..."
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Cited by 11 (4 self)
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We present techniques for implicit solution of discontinuous Galerkin discretizations of the NavierStokes equations on parallel computers. While a blockJacobi method is simple and straightforward to parallelize, its convergence properties are poor except for simple problems. Therefore, we consider NewtonGMRES methods preconditioned with blockincomplete LU factorizations, with optimized element orderings based on a minimum discarded fill (MDF) approach. We discuss the difficulties with the parallelization of these methods, but also show that with a simple domain decomposition approach, most of the advantages of the blockILU over the blockJacobi preconditioner are still retained. The convergence is further improved by incorporating the matrix connectivities into the mesh partitioning process, which aims at minimizing the errors introduced from separating the partitions. We demonstrate the performance of the schemes for realistic two and threedimensional flow problems. I.
An Optimization Framework for Anisotropic Simplex Mesh Adaptation: Application to Aerodynamic Flows
"... We apply an optimizationbased framework for anisotropic simplex mesh adaptation to highorder discontinuous Galerkin discretizations of twodimensional, steadystate aerodynamic flows. The framework iterates toward a mesh that minimizes the output error for a given number of degrees of freedom by c ..."
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Cited by 10 (7 self)
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We apply an optimizationbased framework for anisotropic simplex mesh adaptation to highorder discontinuous Galerkin discretizations of twodimensional, steadystate aerodynamic flows. The framework iterates toward a mesh that minimizes the output error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using elementwise local solves; synthesis of the local errors to construct a surrogate error model in the metric space; and optimization of the surrogate model to drive the mesh toward optimality. The anisotropic adaptation decisions are entirely driven by the behavior of the a posteriori error estimate without making a priori assumptions about the solution behavior. As a result, the method handles any discretization order, naturally incorporates both the primal and adjoint solution behaviors, and robustly treats irregular features. The numerical results demonstrate that the proposed method is at least as competitive as the previous method that relies on a priori assumption of the solution behavior, and, in many cases, outperforms the previous method by over an order of magnitude in terms of the output accuracy for a given number of degrees of freedom. I.
The importance of mesh adaptation for higherorder discretizations of aerodynamic flows
, 2011
"... This work presents an adaptive framework that realizes the true potential of a higherorder discretization of the Reynoldsaveraged NavierStokes (RANS) equations. The framework is based on an outputbased error estimate and explicit degree of freedom control. Adaptation works toward the generation o ..."
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Cited by 10 (6 self)
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This work presents an adaptive framework that realizes the true potential of a higherorder discretization of the Reynoldsaveraged NavierStokes (RANS) equations. The framework is based on an outputbased error estimate and explicit degree of freedom control. Adaptation works toward the generation of meshes that equidistribute local errors and provide anisotropic resolution aligned with solution features in arbitrary orientations. Numerical experiments reveal that uniform refinement limits the performance of higherorder methods when applied to aerodynamic flows with low regularity. However, when combined with aggressive anisotropic refinement of singular features, higherorder methods can significantly improve computational affordability of RANS simulations in the engineering environment. The benefit of the higher spatial accuracy is exhibited for a wide range of applications, including subsonic, transonic, and supersonic flows. The higherorder meshes are generated using the elasticity and the cutcell techniques, and the competitiveness of the cutcell method in terms of accuracy per degree of freedom is demonstrated.
An adaptive shockcapturing HDG method for compressible flows presented at AIAA Conference
, 2011
"... We introduce a hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the compressible Euler equations with shock waves. By locally condensing the approximate conserved variables the HDG method results in a final system involving only the degrees of freedom of the approximate ..."
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Cited by 6 (3 self)
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We introduce a hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the compressible Euler equations with shock waves. By locally condensing the approximate conserved variables the HDG method results in a final system involving only the degrees of freedom of the approximate traces of the conserved variables. The HDG method inherits the geometric flexibility and highorder accuracy of discontinuous Galerkin methods, and offers a significant reduction in the computational cost. In order to treat compressible fluid flows with discontinuities, the HDG method is equipped with an artificial viscosity term based on an extension of existing artificial viscosity methods. Moreover, the artificial viscosity can be used as an indicator for adaptive grid refinement to improve shock profiles. Numerical results for subsonic, transonic, supersonic, and hypersonic flows are presented to demonstrate the performance of the proposed approach. I.
Entropybased Mesh Refinement, I: The entropy adjoint approach
 AIAA paper 20093790, 19th AIAA CFD Meeting
, 2009
"... This work presents a mesh refinement indicator based on entropyvariables, with an application to the compressible NavierStokes equations. The entropy variables are shown to satisfy an adjoint equation, an observation that allows recent work in adjointbased error estimation to be leveraged in con ..."
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Cited by 4 (1 self)
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This work presents a mesh refinement indicator based on entropyvariables, with an application to the compressible NavierStokes equations. The entropy variables are shown to satisfy an adjoint equation, an observation that allows recent work in adjointbased error estimation to be leveraged in constructing a relatively cheap but effective adaptation indicator. The output associated with the entropyvariable adjoint is shown to be the net entropy generation in the computational domain, including physical viscous dissipation when present. Adaptation using entropy variables thus targets areas of the domain responsible for numerical, or spurious, entropy generation. Adaptive results for inviscid and viscous aerodynamic examples demonstrate performance efficiency on par with outputbased adaptation, as measured by errors in various engineering quantities of interest, with the comparative advantage that no adjoint equations need to be solved. I.
HighOrder LES Simulations using ImplicitExplicit RungeKutta schemes, AIAA paper
"... meshes are such that a large number of the elements would allow for explicit timestepping, but the CFLcondition is highly limited by the smaller stretched elements in the boundary layers. We propose an implicitexplicit timeintegration scheme that uses an implicit solver only for the smaller porti ..."
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Cited by 4 (0 self)
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meshes are such that a large number of the elements would allow for explicit timestepping, but the CFLcondition is highly limited by the smaller stretched elements in the boundary layers. We propose an implicitexplicit timeintegration scheme that uses an implicit solver only for the smaller portion of the domain that requires it to avoid severe timestep restrictions, but an efficient explicit solver for the rest of the domain. We use the RungeKutta IMEX schemes and consider several schemes of varying number of stages, orders of accuracy, and stability properties, and study the stability and the accuracy of the solver. We also show the application of the technique on a realistic LEStype problem of turbulent flow around an airfoil, where we conclude that the approach can give performance that is superior to both fully explicit and fully implicit methods. I.
Time implicit highorder discontinuous Galerkin method with reduced evaluation cost
 SIAM J. Sci. Comput
"... Abstract. An efficient and robust time integration procedure for a highorder discontinuous Galerkin method is introduced for solving nonlinear secondorder partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulatio ..."
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Cited by 2 (0 self)
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Abstract. An efficient and robust time integration procedure for a highorder discontinuous Galerkin method is introduced for solving nonlinear secondorder partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between loworder and highorder degrees of freedom. Numerical examples are presented for the nonlinear convectiondiffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A Von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.