A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier–Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier–Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier–Stokes equations are expressed in the convective form and approximated through the Lesaint–Raviart fluxes modified for DG methods.Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier–Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier–Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier–Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.     

Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems

The aim of this paper is to investigate commutative properties of a large family of discontinuous Galerkin (DG) methods applied to optimal control problems governed by the advection-diffusion equations. To compute numerical solutions of PDE constrained optimal control problems there are two main approaches: optimize-then-discretize and discretize-then-optimize. These two approaches do not always coincide and may lead to substantially different numerical solutions. The methods for which these two approaches do coincide we call commutative. In the theory of single equations, there is a related notion of adjoint or dual consistency. In this paper we classify DG methods both in primary and mixed forms and derive necessary conditions that can be used to develop new commutative methods. We will also derive error estimates in the energy and L ² norms. Numerical examples reveal that in the context of PDE constrained optimal control problems a special care needs to be taken to compute the solutions. For example, choosing non-commutative methods and discretize-then-optimize approach may result in a badly behaved numerical solution.     

A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems

In this paper we present a stabilized Discontinuous Galerkin (DG) method for hyperbolic and convection dominated problems. The presented scheme can be used in several space dimension and with a wide range of grid types. The stabilization method preserves the locality of the DG method and therefore allows to apply the same parallelization techniques used for the underlying DG method. As an example problem we consider the Euler equations of gas dynamics for an ideal gas. We demonstrate the stability and accuracy of our method through the detailed study of several test cases in two space dimension on both unstructured and cartesian grids. We show that our stabilization approach preserves the advantages of the DG method in regions where stabilization is not necessary. Furthermore, we give an outlook to adaptive and parallel calculations in 3d.     

Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

In this paper we study explicit peer methods with the strong stability preserving (SSP) property for the numerical solution of hyperbolic conservation laws in one space dimension. A system of ordinary differential equations is obtained by discontinuous Galerkin (DG) spatial discretizations, which are often used in the method of lines approach to solve hyperbolic differential equations. We present in this work the construction of explicit peer methods with stability regions that are designed for DG spatial discretizations and with large SSP coefficients. Methods of second- and third order with up to six stages are optimized with respect to both properties. The methods constructed are tested and compared with appropriate Runge–Kutta methods. The advantage of high stage order is verified numerically.     

Some Recent Developments in Superconvergence of Discontinuous Galerkin Methods for Time-Dependent Partial Differential Equations

In this paper, we briefly review some recent developments in the superconvergence of three types of discontinuous Galerkin (DG) methods for time-dependent partial differential equations: the standard DG method, the local discontinuous Galerkin method, and the direct discontinuous Galerkin method. A survey of our own results for various time-dependent partial differential equations is presented and the superconvergence phenomena of the aforementioned three types of DG solutions are studied for: (i) the function value and derivative approximation at some special points, (ii) cell average error and supercloseness.     

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.     

Interior Penalty Discontinuous Galerkin Method for Maxwell’s Equations in Cold Plasma

In this paper, we develop an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. Both semi and fully discrete DG schemes are constructed, and optimal error estimates in the energy norm are proved. To our best knowledge, this is the first error analysis carried out for the DG method for Maxwell’s equations in dispersive media.     

To CG or to HDG: A Comparative Study

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.     

A CFD-based high-order Discontinuous Galerkin solver for three dimensional electromagnetic scattering problems

In this paper, a CFD (Computational Fluid Dynamics) based DG (Discontinuous Galerkin) method is introduced to solve the three-dimensional Maxwell’s equations for complex geometries on unstructured grids. In order to reduce the computing expense, both the quadrature-free implementation method and the parallel computing based on domain decomposition are employed. On the far-field boundary, the non-reflecting boundary condition is implemented. Numerical integration rather than the quadrature-free implementation is used over the faces on the solid boundary to implement the electromagnetic solid boundary condition for perfectly conducting objectives. Both benchmark examples and complex geometry case are tested with the CFD-based DG solver. Numerical results indicate that highly accurate results can be obtained when using high order even on coarse grid and the present method is very suitable for complex geometries. Furthermore, the costs of CPU time and the speedup of the parallel computation are also evaluated.     

Discontinuous Galerkin Spectral/hp Element Modelling of Dispersive Shallow Water Systems

Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretization in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core

A new atmospheric general circulation model (dynamical core) based on the discontinuous Galerkin (DG) method is developed. This model is conservative, high-order accurate and has been integrated into the NCAR’s high-order method modeling environment (HOMME) to leverage scalable parallel computing capability to thousands of processors. The computational domain for this 3-D hydrostatic model is a cubed-sphere with curvilinear coordinates; the governing equations are cast in flux-form. The horizontal DG discretization employs a high-order nodal basis set of orthogonal Lagrange-Legendre polynomials and fluxes of inter-element boundaries are approximated with Lax-Friedrichs numerical flux. The vertical discretization follows the 1-D vertical Lagrangian coordinates approach combined with the cell-integrated semi-Lagrangian conservative remapping procedure. Time integration follows the third-order strong stability preserving explicit Runge-Kutta scheme. The domain decomposition is applied through space-filling curve approach. To validate the 3-D DG model in HOMME framework, a baroclinic instability test is used and the results are compared with those from the established models. Parallel performance is evaluated on IBM Blue Gene/L supercomputers.     

Connections between the filtered discontinuous Galerkin method and the flux reconstruction approach to high order discretizations

The purpose of this paper is to provide new insights on the connections that exist between the discontinuous Galerkin method (DG), the flux reconstruction method (FR) and the recently identified energy stable flux reconstruction method (ESFR) when solving time dependent conservation laws. All these schemes appear to be quite similar and it is important to understand how they are related. In this paper, we first review results on the stability of the discontinuous Galerkin method and extend it to the filtered discontinuous Galerkin method. We then consider the flux reconstruction approach and show its connections with DG. In particular, we show how the Energy Stable Flux Reconstruction method introduced by Vincent et al. is equivalent to a filtered DG method, hence giving a new proof of its stability. Also, it allows the use of the method without having to know the special form of the flux correction polynomials. Finally, we underline some fundamental differences that exist between FR and DG.     

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.     

Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws

In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method.     

A Discontinuous Galerkin Method for Three-Dimensional Shallow Water Equations

We describe the application of a local discontinuous Galerkin method to the numerical solution of the three-dimensional shallow water equations. The shallow water equations are used to model surface water flows where the hydrostatic pressure assumption is valid. The authors recently developed a DG\linebreak method for the depth-integrated shallow water equations. The method described here is an extension of these ideas to non-depth-integrated models. The method and its implementation are discussed, followed by numerical examples on several test problems.This revised version was published online in July 2005 with corrected volume and issue numbers.     

An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows

This paper presents the latest developments of a discontinuous Galerkin (DG) method for incompressible flows introduced in [Bassi F, Crivellini A, Di Pietro DA, Rebay S. An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J Comput Phys 2006;218(2):794–815] for the steady Navier–Stokes equations and extended in [Bassi F, Crivellini A. A high-order discontinuous Galerkin method for natural convection problems. In: Wesseling P, Oñate E, Periaux J, editors. Electronic proceedings of the ECCOMAS CFD 2006 conference, Egmond aan Zee, The Netherlands, September 5–8; 2006. TU Delft] to the coupled Navier–Stokes and energy equations governing natural convection flows.

The method is fully implicit and applies to the governing equations in primitive variable form. Its distinguishing feature is the formulation of the inviscid interface flux, which is based on the solution of local Riemann problems associated with the artificial compressibility perturbation of the Euler equations. The tight coupling between pressure and velocity so introduced stabilizes the method and allows using equal-order approximation spaces for both pressure and velocity. Since, independently of the amount of artificial compressibility added, the interface flux reduces to the physical one for vanishing interface jumps, the resulting method is strongly consistent.

In this paper, we present a review of the method together with two recently developed issues: (i) the high-order DG discretization of the incompressible Euler equations; (ii) the high-order implicit time integration of unsteady flows. The accuracy and versatility of the method are demonstrated by a suite of computations of steady and unsteady, inviscid and viscous incompressible flows.     

A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations

In this work we consider Runge–Kutta discontinuous Galerkin methods for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.     

Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.This revised version was published online in July 2005 with corrected volume and issue numbers.     

A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.     

A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method.