A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss–Lobatto–Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge–Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400–5000 and flows around an impulsively started cylinder for Re = 550–9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.     

Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations on unstructured meshes. The algorithms are based on coupling both p- and h-multigrid (ph-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton–Krylov method.The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton–GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem.It is concluded that the Newton–GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.     

Artificial boundary conditions for the numerical solution of the Euler equations by the discontinuous galerkin method

We present artificial boundary conditions for the numerical simulation of compressible flows using high-order accurate discretizations with the discontinuous Galerkin (DG) finite element method. The construction of the proposed boundary conditions is based on characteristic analysis and applied for boundaries with arbitrary shape and orientation. Numerical experiments demonstrate that the proposed boundary treatment enables to convect out of the computational domain complex flow features with little distortion. In addition, it is shown that small-amplitude acoustic disturbances could be convected out of the computational domain, with no significant deterioration of the overall accuracy of the method. Furthermore, it was found that application of the proposed boundary treatment for viscous flow over a cylinder yields superior performance compared to simple extrapolation methods.     

hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space–time discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge–Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space–time discontinuous Galerkin discretization of the advection–diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for steady state and time-dependent problems, and low and high cell Reynolds numbers, including highly stretched meshes.     

A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.     

Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors

We consider a family of explicit one-step time discretizations for finite volume and discontinuous Galerkin schemes, which is based on a predictor-corrector formulation. The predictor remains local taking into account the time evolution of the data only within the grid cell. Based on a space–time Taylor expansion, this idea is already inherent in the MUSCL finite volume scheme to get second order accuracy in time and was generalized in the context of higher order ENO finite volume schemes. We interpret the space–time Taylor expansion used in this approach as a local predictor and conclude that other space–time approximate solutions of the local Cauchy problem in the grid cell may be applied. Three possibilities are considered in this paper: (1) the classical space–time Taylor expansion, in which time derivatives are obtained from known space-derivatives by the Cauchy–Kovalewsky procedure; (2) a local continuous extension Runge–Kutta scheme and (3) a local space–time Galerkin predictor with a version suitable for stiff source terms. The advantage of the predictor–corrector formulation is that the time evolution is done in one step which establishes optimal locality during the whole time step. This time discretization scheme can be used within all schemes which are based on a piecewise continuous approximation as finite volume schemes, discontinuous Galerkin schemes or the recently proposed reconstructed discontinuous Galerkin or P_NP_M schemes. The implementation of these approaches is described, advantages and disadvantages of different predictors are discussed and numerical results are shown.     

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.     

Uncertainty quantification for systems of conservation laws

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. In this paper, we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with Polynomial Chaos (PC) methods. The idea is to introduce a new variable, the entropic variable, in bijection with our vector of unknowns, which we develop on the polynomial basis: by performing a Galerkin projection, we obtain a deterministic system of conservation laws. We state several properties of this deterministic system in the case of a general uncertain system of conservation laws. We then apply the method to the case of the inviscid Burgers’ equation with random initial conditions and we present some preliminary results for the Euler system. We systematically compare results from our new approach to results from the stochastic Galerkin method. In the vicinity of discontinuities, the new method bounds the oscillations due to Gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations. It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.     

A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods

We present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method.     

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.     

A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions

Standard Galerkin discretization techniques (with locally- or globally-supported basis functions) for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension, leading to large CPU time and memory requirements to set up the dense Galerkin matrix, with each entry requiring evaluation of multi-dimensional highly oscillatory integrals. In this work, using globally-supported basis functions, we describe an efficient fully discrete Galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional Galerkin integrals with high-order accuracy. Such high-order order accuracy for various practically relevant frequencies (k ∈ [1, 100,000]) substantially improves on approximations based on standard asymptotic techniques. We demonstrate the efficiency of our algorithm for spherical and non-spherical convex scattering for several wavenumbers 1 ? k ? 100,000 for low to high order prescribed tolerance. Our fully discrete algorithm requires only mild growth in the number of unknowns and CPU time as the frequency increases.     

二维Lagrangian坐标系下可压气动方程组的间断Petrov-Galerkin方法

构造矩形网格下求解Lagrangian坐标系下气动方程组的单元中心型格式. 空间离散采用控制体积间断Petrov-Galerkin方法,时间离散采用二阶TVD Runge-Kutta方法. 利用限制器来抑制非物理震荡并保证RKCV算法的稳定性. 构造的算法可以保证物理量的局部守恒. 与Runge-Kutta间断Galerkin（RKDG）方法相比较,RKCV方法的计算公式少一项积分项使得计算较简单. 给出一些数值算例验证了算法的可靠性及效率.     

Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection–diffusion equation

An improved p-multigrid algorithm for discontinuous Galerkin (DG) discretizations of convection–diffusion problems is presented. The general p  -multigrid algorithm for DG discretizations involves a restriction from the p=1$p=1$ to p=0$p=0$ discontinuous polynomial solution spaces. This restriction is problematic and has limited the efficiency of the p  -multigrid method. For purely diffusive problems, Helenbrook and Atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p=1$p=1$. It is shown that this method is not directly applicable to the convection–diffusion (CD) equation because it results in a central-difference discretization for the convective term. To remedy this, ideas from the streamwise upwind Petrov–Galerkin (SUPG) formulation are used to devise a transition from the discontinuous to continuous space at p=1$p=1$ that yields an upwind discretization. The results show that the new method converges rapidly for all Peclet numbers.     

Polymorphic nodal elements and their application in discontinuous Galerkin methods

In this work, we discuss two different but related aspects of the development of efficient discontinuous Galerkin methods on hybrid element grids for the computational modeling of gas dynamics in complex geometries or with adapted grids. In the first part, a recursive construction of different nodal sets for hp finite elements is presented. They share the property that the nodes along the sides of the two-dimensional elements and along the edges of the three-dimensional elements are the Legendre–Gauss–Lobatto points. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. In the second part, these nodal elements are applied within the modal discontinuous Galerkin framework. We still use a modal based formulation, but introduce a nodal based integration technique to reduce computational cost in the spirit of pseudospectral methods. We illustrate the performance of the scheme on several large scale applications and discuss its use in a recently developed space-time expansion discontinuous Galerkin scheme.     

Preconditioning for dual-time-stepping simulations of the shallow water equations including Coriolis and bed friction effects

Diagonal preconditioners for implicit-unsteady and steady discretizations of the shallow water equations at low- and high-Rossby and drag-number limits are analyzed. For each case, sub and supercritical flow conditions are also considered. Based on the analysis, a preconditioner is derived for use with a multigrid cycle that performs as well as possible under all conditions. In addition, a streamwise-upwind Petrov–Galerkin discretization of the system is presented that is derived from the preconditioned system. Using this discretization, it is demonstrated that for most conditions, the preconditioner gives rapid convergence that is independent of the grid resolution and the flow parameters. Practical tests including an equatorial Rossby soliton and tide propagation over variable bathymetry are simulated to demonstrate the performance of this approach.     

高阶间断伽辽金时域有限元方法分析三维谐振腔

从一阶麦克斯韦旋度方程出发,研究一种区域分解时域有限元目的——高阶间断伽辽金时域有限元目的.其中对时间的离散采用Crank-Nicolson差分格式,电场和磁场采用相同阶数的高阶矢量基函数展开.分析三维谐振腔问题,数值结果表明,目的 中时间步长的选取可以摆脱CFL稳定性条件的限制;此外,与基于常用Whitney矢量基函数的目的 相比,采用高阶矢量基函数可以明显地提高计算精度及计算效率.     

Preconditioning methods for discontinuous Galerkin solutions of the Navier–Stokes equations

A Newton–Krylov method is developed for the solution of the steady compressible Navier–Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton–Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill–Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.     

LWDG Method for a Multi-Class Traffic Flow Model on an Inhomogeneous Highway

In this paper, we apply the discontinuous Galerkin method with Lax-Wendroff type time discretizations (LWDG) using the weighted essentially non-oscillatory (WENO) limiter to solve a multi-class traffic flow model for an inhomogeneous highway, which is a kind of hyperbolic conservation law with spatially varying fluxes. The numerical scheme is based on a modified equivalent system which is written as a "standard" hyperbolic conservation form. Numerical experiments for both the Riemann problem and the traffic signal control problem are presented to show the effectiveness of these methods.     

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.