A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier–Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric modeling. We study three different forms of the governing equations using seven test cases. Three test cases involve flow over mountains which require the implementation of non-reflecting boundary conditions, while one test requires viscous terms (density current). Including viscous stresses into finite difference, finite element, or spectral element models poses no additional challenges; however, including these terms to either finite volume or discontinuous Galerkin models requires the introduction of additional machinery because these methods were originally designed for first-order operators. We use the local discontinuous Galerkin method to overcome this obstacle. The seven test cases show that all of our models yield good results. The main conclusion is that equation set 1 (non-conservation form) does not perform as well as sets 2 and 3 (conservation forms). For the density current (viscous), the SE and DG models using set 3 (mass and total energy) give less dissipative results than the other equation sets; based on these results we recommend set 3 for the development of future multiscale research codes. In addition, the fact that set 3 conserves both mass and energy up to machine precision motives us to pursue this equation set for the development of future mesoscale models. For the bubble and mountain tests, the DG models performed better. Based on these results and due to its conservation properties we recommend the DG method. In the worst case scenario, the DG models are 50% slower than the non-conservative SE models. In the best case scenario, the DG models are just as efficient as the conservative SE models.     

Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations

A new type of ultra-high resolution atmospheric global circulation model is developed. The new model is designed to perform “cloud resolving simulations” by directly calculating deep convection and meso-scale circulations, which play key roles not only in the tropical circulations but in the global circulations of the atmosphere. Since cores of deep convection have a few km in horizontal size, they have not directly been resolved by existing atmospheric general circulation models (AGCMs). In order to drastically enhance horizontal resolution, a new framework of a global atmospheric model is required; we adopted nonhydrostatic governing equations and icosahedral grids to the new model, and call it Nonhydrostatic ICosahedral Atmospheric Model (NICAM). In this article, we review governing equations and numerical techniques employed, and present the results from the unique 3.5-km mesh global experiments—with O(10⁹) computational nodes—using realistic topography and land/ocean surface thermal forcing. The results show realistic behaviors of multi-scale convective systems in the tropics, which have not been captured by AGCMs. We also argue future perspective of the roles of the new model in the next generation atmospheric sciences.     

The Elastoplast Discontinuous Galerkin (EDG) method for the Navier–Stokes equations

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 L² 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.     

Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations

This article considers a posteriori error estimation and anisotropic mesh refinement for three-dimensional laminar aerodynamic flow simulations. The optimal order symmetric interior penalty discontinuous Galerkin discretization which has previously been developed for the compressible Navier–Stokes equations in two dimensions is extended to three dimensions. Symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model. Using duality arguments, an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients. Furthermore, residual-based indicators as well as adjoint-based indicators for goal-oriented refinement are derived. These refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous Galerkin (DG) discretization. Two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint-based error estimation are presented. The performance of the proposed discretization, error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows.     

A non-uniform basis order for the discontinuous Galerkin method of the 3D dissipative wave equation with perfectly matched layer

In this study a discontinuous Galerkin method (DG) for solving the three-dimensional time-dependent dissipative wave equation is investigated. In the case of unbounded problems, the perfectly matching layer (PML) is used to truncate the computational domain. The aim of this work is to investigate a simple selection method for choosing the basis order for elements in the computational mesh in order to obtain a predetermined error level. The selection method studied here relies on the error estimates provided by Ainsworth [M. Ainsworth, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, Journal of Computational Physics 198(1) (2004) 106–130]. The performance of the non-uniform basis is examined using numerical experiments. In the simulated model problems, a feasible method choosing the basis order for arbitrary sized elements is achieved. In simulations, the effect of dissipation and the choices of the PML parameters on the performance of the DG method are also investigated.     

基于静动态重构的高阶DG/FV混合格式在二维非结构网格中的推广

通过比较间断Galerkin有限元方法(DGM)和有限体积方法(FVM),提出"静态重构"和"动态重构"的概念,进一步建立基于静动态"混合重构"算法的三阶DG/FV混合格式.在DG/FV混合格式中,单元平均值和一阶导数由DGM方法"动态重构",二阶导数利用FVM方法"静态重构";在此基础上,构造高阶多项式插值函数,得到...     

A Discontinuous Galerkin Method Based on a BGK Scheme for the Navier-Stokes Equations on Arbitrary Grids

A discontinuous Galerkin Method based on a Bhatnagar-Gross-Krook (BGK) formulation is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The idea behind this approach is to combine the robustness of the BGK scheme with the accuracy of the DG methods in an effort to develop a more accurate, efficient, and robust method for numerical simulations of viscous flows in a wide range of flow regimes. Unlike the traditional discontinuous Galerkin methods, where a Local Discontinuous Galerkin (LDG) formulation is usually used to discretize the viscous fluxes in the Navier-Stokes equations, this DG method uses a BGK scheme to compute the fluxes which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. The developed method is used to compute a variety of viscous flow problems on arbitrary grids. The numerical results obtained by this BGKDG method are extremely promising and encouraging in terms of both accuracy and robustness, indicating its ability and potential to become not just a competitive but simply a superior approach than the current available numerical methods.     

Modeling and numerical approximation of a 2.5D set of equations for mesoscale atmospheric processes

The set of 3D inviscid primitive equations for the atmosphere is dimensionally reduced by a Discontinuous Galerkin discretization in one horizontal direction. The resulting model is a 2D system of balance laws with a source term depending on the layering procedure and the choice of coupling fluxes, which is established in terms of upwind considerations. The “2.5D” system is discretized via a WENO–TVD scheme based in a flux-limiter approach. We study four tests cases related to atmospheric phenomena to analyze the physical validity of the model.     

Three-dimensional electromagnetic band-gap structure containing TiO2 fabricated by rapid-prototyping

Three-dimensional (3D) diamond structure electromagnetic band-gap (EBG) structures containing TiO₂ fabricated by rapid-prototyping (RP) technique were investigated. The simulations based on finite element method (FEM) were employed to model the band structures. The influence of aspect ratio on the band gap width was studied. The optimal band gap width EBGs were fabricated and investigated experimentally. Gel-casting together with RP technique were used in the fabrication. TiO₂ gel was cast into the diamond structure molds fabricated by RP method to obtain the green EBG structures. The transmission characteristics of the EBG structures were measured by transmission/reflection (T/R) methods using a vector network analyzer. Complete band-gap was observed in the transmission characteristics in the frequency from 11 GHz to 12 GHz, which agreed well with the simulation results.     

Free-space laser communication performance in the atmospheric channel

In spite of the tremendous technical advancement of available components, the major limitation of free-space laser communication (lasercom) performance is due to the atmosphere, because a portion of the atmospheric path always includes turbulence and multiple scattering effects. Starting from a fundamental understanding of the laser communications system under diverse weather conditions, this chapter provides a comprehensive treatment of the evaluation of parameters needed for analyzing system performance. The significance of higher-order statistics of probability density functions of irradiance fluctuations due to turbulence to performance analysis is explained. Starting from link analysis, the necessary expressions relating link margin, bit-error-rate, signal-to-noise-ratio, and probability of fade statistics are presented. Results for laboratory-simulated atmospheric turbulence and multiple scattering are presented. Example numerical results for simulations of lasercom systems operating under various atmospheric conditions are presented for various scenarios such as uplink-downlink (e.g., between ground and satellite, aircraft or UAV) and horizontal (terrestrial) link. Both turbulence and multiple scattering effects have been included in the analysis with both on-off keying and pulse-position modulation schemes. Statistical estimation and computation of communication parameters presented in this chapter will be useful in designing and optimizing lasercom systems that are reliable under all weather conditions.     

基于自适应光学补偿的自由空间光通信系统性能研究

自由空间光通信在未来全球通信网中具有潜在的应用价值,然而自由空间光通信的性能易受大气湍流影响,而自适应光学能够解决大气湍流问题.在gamma-gamma分布大气湍流中,用自适应光学技术对自由空间光通信系统进行补偿,进行通信系统误码率的分析,给出了上行链路和下行链路的模拟结果.结果表明:自适应光学误差补偿技术能够很好地提高自由空间光通信的性能,并且低阶自适应光学补偿就能达到很好的校正效果.     

非静力模式中三维网格的计算特性

在线性斜压非静力滞弹性方程组的基础上,从频率和群速方面讨论了10种三维网格的计算频散性,结果表明三维网格EL/CP、C/CP与Z/LZ计算频散性能较好.从而为非静力原始方程大气模式选取三维网格提供指导.     

GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model

The compute unified device architecture (CUDA) is a programming approach for performing scientific calculations on a graphics processing unit (GPU) as a data-parallel computing device. The programming interface allows to implement algorithms using extensions to standard C language. With continuously increased number of cores in combination with a high memory bandwidth, a recent GPU offers incredible resources for general purpose computing. First, we apply this new technology to Monte Carlo simulations of the two dimensional ferromagnetic square lattice Ising model. By implementing a variant of the checkerboard algorithm, results are obtained up to 60 times faster on the GPU than on a current CPU core. An implementation of the three dimensional ferromagnetic cubic lattice Ising model on a GPU is able to generate results up to 35 times faster than on a current CPU core. As proof of concept we calculate the critical temperature of the 2D and 3D Ising model using finite size scaling techniques. Theoretical results for the 2D Ising model and previous simulation results for the 3D Ising model can be reproduced.     

Application of Galerkin spectral method for tearing mode instability

Magnetic reconnection and tearing mode instability play a critical role in many physical processes. The application of Galerkin spectral method for tearing mode instability in two-dimensional geometry is investigated in this paper. A resistive magnetohydrodynamic code is developed, by the Galerkin spectral method both in the periodic and aperiodic directions. Spectral schemes are provided for global modes and local modes. Mode structures, resistivity scaling, convergence and stability of tearing modes are discussed. The effectiveness of the code is demonstrated, and the computational results are compared with the results using Galerkin spectral method only in the periodic direction. The numerical results show that the code using Galerkin spectral method individually allows larger time step in global and local modes simulations, and has better convergence in global modes simulations.     

Numerical simulation of non-viscous liquid pinch-off using a coupled level set-boundary integral method

Simulations of the pinch-off of an inviscid fluid column are carried out based upon a potential flow model with capillary forces. The interface location and the time evolution of the free surface boundary condition are both approximated by means of level set techniques on a fixed domain. The interface velocity is obtained via a Galerkin boundary integral solution of the 3D axisymmetric Laplace equation. A short-time analytical solution of the Raleigh–Taylor instability in a liquid column is available, and this result is compared with our numerical experiments to validate the algorithm. The method is capable of handling pinch-off and after pinch-off events, and simulations showing the time evolution of the fluid tube are presented.     

Iterative regularization methods for atmospheric remote sensing

In this paper we present different inversion algorithms for nonlinear ill-posed problems arising in atmosphere remote sensing. The proposed methods are Landweber's method (LwM), the iteratively regularized Gauss-Newton method, and the conventional and regularizing Levenberg-Marquardt method. In addition, some accelerated LwMs and a technique for smoothing the Levenberg-Marquardt solution are proposed. The numerical performance of the methods is studied by means of simulations. Results are presented for an inverse problem in atmospheric remote sensing, i.e., temperature sounding with an airborne uplooking high-resolution far-infrared spectrometer.     

An analysis of time discretization in the finite element solution of hyperbolic problems

The problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined. A modified equation analysis reveals that the classical, second-order accurate, time-stepping algorithms, i.e., the Lax-Wendroff, leap-frog, and Crank-Nicolson methods, properly combine with piecewise linear finite elements in advection problems only for small values of the time step. On the contrary, as the Courant number increases, the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit CFL property are not achieved. These fundamental numerical properties can, however, be recovered, while still remaining in the standard Galerkin finite element setting, by increasing the order of accuracy of the time discretization. This is accomplished by exploiting the Taylor series expansion in the time increment up to the third order before performing the Galerkin spatial discretization using piecewise linear interpolations. As a result, it appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods). Numerical results for several linear examples are presented to illustrate the properties of the Taylor-Galerkin schemes in one- and two-dimensional calculations.     

A time-domain Discontinuous Galerkin method for mechanical resonator quality factor computations

Numerical simulations are becoming increasingly important in the design of micromechanical resonators, in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses. Frequency based approaches have been shown to predict these accurately, however, they require the solution of eigenvalue problems or the inversion of Helmholtz-type operators which are known to be very difficult for large-scale iterative solvers. We propose using a time-domain approach instead, where a broadband input signal is propagated through the system with a local explicit time-stepper. This is achieved using a new high-order Discontinuous Galerkin (DG) discretization for the linear elasticity equations, in particular a second-order formulation with Compact DG fluxes and a Runge–Kutta time integrator, where the block-diagonal mass matrices allow for efficient, stable, and accurate time stepping. Our solver scales well on distributed parallel computers, even in three spatial dimension and for large problem sizes. The resulting output signal is analyzed using a well-known filter diagonalization method, which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data. We validate the properties of our scheme on model problems, and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators, using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well.     

湍流大气光通信系统误码率分析与实验研究

自由空间光通信(FSO)系统的性能由于受大气湍流影响会产生剧烈波动。根据系统和大气参数评估系统差错性能的研究具有现实意义。以大气湍流信道和光电探测两个模型为基础,建立了FSO系统差错性能的数学仿真模型,提出了湍流条件下系统误码率计算公式。对仿真结果与弱湍流条件下获得的实验数据进行了比较,并依据此模型对光强起伏和背景噪声等因素的影响进行仿真。仿真结果表明,基于该模型的仿真结果与实验数据一致,光强起伏是引起系统性能波动的主要因素,最优判决阈值需根据实际大气条件进行调整。该模型可有效评估湍流条件下FSO系统性能,并为相关理论研究提供参考。