Ngoc Cuong Nguyen

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection-diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton-Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is (p+1)th order accurate and when polynomials of degree p>=0 are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p+1 in the L^2-norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p+1 in the L^2-norm. The new approximate scalar variable is shown to converge with order p+2 in the L^2-norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.

An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier-Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure-velocity formulation of the incompressible Navier-Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k+1 in the L2-norm, when polynomials of degree k?0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H(div)-conforming, and converges with order k+2 for k?1 and with order 1 for k=0 in the L2-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.

RANS Solutions Using High Order Discontinuous Galerkin Methods

We present a practical approach for the numerical solution of the Reynolds averaged Navier-Stokes (RANS) equations using high-order discontinuous Galerkin methods. Turbulence is modeled by the Spalart-Allmaras (SA) one-equation model. We introduce an artificial viscosity model for SA equation which is aimed at accommodating high-order RANS approximations on grids which would otherwise be too coarse. Generally, the model term is only active at the edge of the boundary layer, where the grid resolution is insufficient to capture the abrupt change in curvature required for the eddy viscosity profile to match its free-stream value. Furthermore, the amount of viscosity required decreases with the grid resolution and vanishes when the resolution is sufficiently high. For transonic computations, an additional shock-capturing artificial viscosity model term is required. Numerical predictions for turbulent flows past a flat plate and a NACA 0012 airfoil are presented via comparison with the experimental measurements. In the flat plate case, grid refinement studies are performed in order to assess the convergence properties and demonstrate the effectiveness of high-order approximations.

High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics

We present a class of hybridizable discontinuous Galerkin (HDG) methods for the numerical simulation of wave phenomena in acoustics and elastodynamics. The methods are fully implicit and high-order accurate in both space and time, yet computationally attractive owing to their following distinctive features. First, they reduce the globally coupled unknowns to the approximate trace of the velocity, which is defined on the element faces and single-valued, thereby leading to a significant saving in the computational cost. In addition, all the approximate variables (including the approximate velocity and gradient) converge with the optimal order of k+1 in the L^2-norm, when polynomials of degree k>=0 are used to represent the numerical solution and when the time-stepping method is accurate with order k+1. When the time-stepping method is of order k+2, superconvergence properties allows us, by means of local postprocessing, to obtain better, yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k+2 in the L^2-norm when k>=1 for both acoustics and elastodynamics. Extensive numerical results are provided to illustrate these distinctive features.

Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations

We present two hybridizable discontinuous Galerkin (HDG) methods for the numerical solution of the time-harmonic Maxwells equations. The first HDG method explicitly enforces the divergence-free condition and thus necessitates the introduction of a Lagrange multiplier. It produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the Lagrange multiplier. The second HDG method does not explicitly enforce the divergence-free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only. For both HDG methods, the approximate vector field converges with the optimal order of k+1 in the L^2-norm, when polynomials of degree k are used to represent all the approximate variables. We propose elementwise postprocessing to obtain a new H^c^u^r^l-conforming approximate vector field which converges with order k+1 in the H^c^u^r^l-norm. We present extensive numerical examples to demonstrate and compare the performance of the HDG methods.

Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics

We present hybridizable discontinuous Galerkin methods for solving steady and time-dependent partial differential equations (PDEs) in continuum mechanics. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the numerical flux to provide stability and consistency; and a global jump condition that enforces the continuity of the numerical flux to arrive at a global weak formulation in terms of the numerical trace. The HDG methods are fully implicit, high-order accurate and endowed with several unique features which distinguish themselves from other discontinuous Galerkin methods. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, they provide, for smooth viscous-dominated problems, approximations of all the variables which converge with the optimal order of k+1 in the L2-norm. Third, they possess some superconvergence properties that allow us to define inexpensive element-by-element postprocessing procedures to compute a new approximate solution which may converge with higher order than the original solution. And fourth, they allow for a novel and systematic way for imposing boundary conditions for the total stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the methods. In addition, they possess other interesting properties for specific problems. Their approximate solution can be postprocessed to yield an exactly divergence-free and H(div)-conforming velocity field for incompressible flows. They do not exhibit volumetric locking for nearly incompressible solids. We provide extensive numerical results to illustrate their distinct characteristics and compare their performance with that of continuous Galerkin methods.

Hybridizable Discontinuous Galerkin Methods

We present an overview of recent developments of HDG methods for numerically solving partial differential equations in fluid mechanics.

A Comparison of HDG Methods for Stokes Flow

In this paper, we compare hybridizable discontinuous Galerkin (HDG) methods for numerically solving the velocity-pressure-gradient, velocity-pressure-stress, and velocity-pressure-vorticity formulations of Stokes flow. Although they are defined by using different formulations of the Stokes equations, the methods share several common features. First, they use polynomials of degree k for all the components of the approximate solution. Second, they have the same globally coupled variables, namely, the approximate trace of the velocity on the faces and the mean of the pressure on the elements. Third, they give rise to a matrix system of the same size, sparsity structure and similar condition number. As a result, they have the same computational complexity and storage requirement. And fourth, they can provide, by means of an element-by element postprocessing, a new approximation of the velocity which, unlike the original velocity, is divergence-free and H(div)-conforming. We present numerical results showing that each of the approximations provided by these three methods converge with the optimal order of k+1 in L2 for any k≥0. We also display experiments indicating that the postprocessed velocity is a better approximation than the original approximate velocity. It converges with an additional order than the original velocity for the gradient-based HDG, and with the same order for the vorticity-based HDG methods. For the stress-based HDG methods, it seems to converge with an additional order for even polynomial degree approximations. Finally, the numerical results indicate that the method based on the velocity-pressure-gradient formulation provides the best approximations for similar computational complexity.

Navier-Stokes Solution Using Hybridizable Discontinuous Galerkin methods

We are concerned with the numerical solution of the Navier-Stokes and Reynoldsaveraged Navier-Stokes equations using the Hybridizable Discontinuous Galerkin (HDG) methods recently introduced in Ref. [34]. These methods are computationally more ecient and accurate than other discontinuous Galerkin methods and hence, well suited to be applied to CFD problems. However, in order for them to be able to deal with the range of problems of relevance to Aeronautics, both turbulence and shocks have to be properly addressed. First, we will present a modication of the Spalart-Allmaras (SA) turbulence model that improves the convergence properties of the method by means of a regularization of the working variable; this modication is eective only in regions where the eddy viscosity is smaller than the molecular viscosity, therefore, it does not aect the numerical prediction of ow quantities as compared to the original SA model. Then, an articial viscosity coecient driven by the divergence of the velocity will be implemented in order to deal with shock waves. Numerical results are presented to demonstrate the proposed approach in several instances, from laminar separated ows to turbulent compressible ows.

An Embedded Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations

We present an Embedded Discontinuous Galerkin (EDG) 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 uxes and weakly imposing the continuity of the normal component of the numerical uxes across the element interfaces. This allows the approximate conserved variables dening 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 EDG method can be seen as a particular form of a Hybridizable Discontinuous Galerkin (HDG) method in which the hybrid uxes are required to belong to a smaller space than in standard HDG methods. In our EDG method, the hybrid unknown is taken to be continuous at the vertices, thus resulting in an even smaller number of coupled degrees of freedom than in the HDG method. In fact, the resulting stiness matrix has the same structure as that of the statically condensed continuous Galerkin method. In exchange for the reduced number of degrees of freedom, the EDG method looses the optimal converge property of the ux which characterizes other HDG methods. Thus, for convection-diusi on problems, the EDG solution converges optimally for the primal unknown but suboptimally for the ux.