An Embedded Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations (original) (raw)
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A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes 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.
A class of embedded discontinuous Galerkin methods for computational fluid dynamics
Journal of Computational Physics, 2015
We present a class of embedded discontinuous Galerkin (EDG) methods for numerically solving the Euler equations and the Navier-Stokes equations. The essential ingredients are a local Galerkin projection of the underlying governing equations at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the approximate trace, a judicious choice of the numerical flux to provide stability and consistency, and a global jump condition that weakly enforces the single-valuedness of the numerical flux to arrive at a global formulation in terms of the numerical trace. The EDG methods are thus obtained from the hybridizable discontinuous Galerkin (HDG) method by requiring the approximate trace to belong to smaller approximation spaces than the one in the HDG method. In the EDG methods, the numerical trace is taken to be continuous on a suitable collection of faces, thus resulting in an even smaller number of globally coupled degrees of freedom than in the HDG method. On the other hand, the EDG methods are no longer locally conservative. In the framework of convection-diffusion problems, this lack of local conservativity is reflected in the fact that the EDG methods do not provide the optimal convergence of the approximate gradient or the superconvergence for the scalar variable for diffusion-dominated problems as the HDG method does. However, since the HDG method does not display these properties in the convection-dominated regime, the EDG method becomes a reasonable alternative since it produces smaller algebraic systems than the HDG method. In fact, the resulting stiffness matrix has a similar sparsity pattern as that of the statically condensed continuous Galerkin (CG) method. The main advantage of the EDG methods is that they are generally more stable and robust than the CG method for solving convection-dominated problems. Numerical results are presented to illustrate the performance of the EDG methods. They confirm that, even though the EDG methods are not locally conservative, they are a viable alternative to the HDG method in the convectiondominated regime.
A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems
2009
In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes 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.
arXiv (Cornell University), 2024
The macro-element variant of the hybridized discontinuous Galerkin (HDG) method combines advantages of continuous and discontinuous finite element discretization. In this paper, we investigate the performance of the macro-element HDG method for the analysis of compressible flow problems at moderate Reynolds numbers. To efficiently handle the corresponding large systems of equations, we explore several strategies at the solver level. On the one hand, we devise a second-layer static condensation approach that reduces the size of the local system matrix in each macroelement and hence the factorization time of the local solver. On the other hand, we employ a multi-level preconditioner based on the FGMRES solver for the global system that integrates well within a matrix-free implementation. In addition, we integrate a standard diagonally implicit Runge-Kutta scheme for time integration. We test the matrix-free macro-element HDG method for compressible flow benchmarks, including Couette flow, flow past a sphere, and the Taylor-Green vortex. Our results show that unlike standard HDG, the macro-element HDG method can operate efficiently for moderate polynomial degrees, as the local computational load can be flexibly increased via mesh refinement within a macro-element. Our results also show that due to the balance of local and global operations, the reduction in degrees of freedom, and the reduction of the global problem size and the number of iterations for its solution, the macro-element HDG method can be a competitive option for the analysis of compressible flow problems.
In the spirit of making high-order discontinuous Galerkin (DG) methods more competitive, researchers have developed the hybridized DG methods, a class of discontinuous Galerkin methods that generalizes the Hybridizable DG (HDG), the Embedded DG (EDG) and the Interior Embedded DG (IEDG) methods. These methods are amenable to hybridization (static condensation) and thus to more computationally efficient implementations. Like other high-order DG methods, however, they may suffer from numerical stability issues in under-resolved fluid flow simulations. In this spirit, we introduce the hybridized DG methods for the compressible Euler and Navier-Stokes equations in entropy variables. Under a suitable choice of the numerical flux, the scheme can be shown to be entropy stable and satisfy the Second Law of Thermodynamics in an integral sense. The performance and robustness of the proposed family of schemes are illustrated through a series of steady and unsteady flow problems in subsonic, transonic, and supersonic regimes. The hybridized DG methods in entropy variables show the optimal accuracy order given by the polynomial approximation space, and are significantly superior to their counterparts in conservation variables in terms of stability and robustness, particularly for under-resolved and shock flows.
Hybridizable Discontinuous Galerkin Methods
Lecture notes in computational science and engineering, 2010
In this paper, we present and discuss the so-called hybridizable discontinuous Galerkin (HDG) methods. The discontinuous Galerkin (DG) methods were originally devised for numerically solving linear and then nonlinear hyperbolic problems. Their success prompted their extension to the compressible Navier-Stokes equations-and hence to second-order elliptic equations. The clash between the DG methods and decades-old, well-established finite element methods resulted in the introduction of the HDG methods. The HDG methods can be implemented more e ciently and are more accurate than all previously known DG methods; they represent a competitive alternative to the well established finite element methods. Here we show how to devise and implement the HDG methods, argue why they work so well and prove optimal convergence properties in the framework of di↵usion and incompressible flow problems. We end by briefly describing extensions to other continuum mechanics and fluid dynamics problems.
A hybridizable discontinuous Galerkin method for Stokes flow
Computer Methods in Applied Mechanics and Engineering, 2010
In this paper, we introduce a hybridizable discontinuous Galerkin method for Stokes flow. The method is devised by using the discontinuous Galerkin methodology to discretize a velocity-pressure-gradient formulation of the Stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization. One of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries, thereby leading to a significant reduction in the size of the resulting matrix. Moreover, by using an augmented lagrangian method, the globally coupled unknowns are further reduced to the numerical trace of the velocity only. Another important feature is that the approximations of the velocity, pressure, and gradient converge with the optimal order of k þ 1 in the L 2-norm, when polynomials of degree k P 0 are used to represent the approximate variables. Based on the optimal convergence of the HDG method, we apply an element-by-element postprocessing scheme to obtain a new approximate velocity, which converges with order k þ 2 in the L 2-norm for k P 1. The postprocessing performed at the element level is less expensive than the solution procedure. Numerical results are provided to assess the performance of the method.
The Elastoplast Discontinuous Galerkin (EDG) method for the Navier–Stokes equations
Journal of Computational Physics, 2012
The present work details the Elastoplast (this name is a translation from the French ''sparadrap'', a concept first applied by Yves Morchoisne for Spectral methods [1]) Discontinuous Galerkin (EDG) method to solve the compressible Navier-Stokes equations. This method was first presented in 2009 at the ICOSAHOM congress with some Cartesian grid applications. We focus here on unstructured grid applications for which the EDG method seems very attractive. As in the Recovery method presented by van Leer and Nomura in 2005 for diffusion, jumps across element boundaries are locally eliminated by recovering the solution on an overlapping cell. In the case of Recovery, this cell is the union of the two neighboring cells and the Galerkin basis is twice as large as the basis used for one element. In our proposed method the solution is rebuilt through an L 2 projection of the discontinuous interface solution on a small rectangular overlapping interface element, named Elastoplast, with an orthogonal basis of the same order as the one in the neighboring cells. Comparisons on 1D and 2D scalar diffusion problems in terms of accuracy and stability with other viscous DG schemes are first given. Then, 2D results on acoustic problems, vortex problems and boundary layer problems both on Cartesian or unstructured triangular grids illustrate stability, precision and versatility of this method.
An implicit discontinuous Galerkin method for the unsteady compressible Navier–Stokes equations
Computers & Fluids, 2012
A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier-Stokes equations. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A second order implicit method is applied for temporal discretization to the resulting ordinary differential equations. The resulting nonlinear system of equations is solved at each time step using a pseudo-time marching approach. A newly developed fast, p-multigrid is then used to obtain the steady state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to orders of improvements in performance over its explicit counterpart, while without significant increase in memory requirements.