非定常喷流有限差分方法数值试验

以欠膨胀自由喷流初期流动为例,采用Euler方程和Beam Warming TVD有限差分格式,对比分析了Beam WarmingAF方法,隐式亚迭代方法和简化Runge Kutta五步格式的非定常流场描述能力,结果表明:(1)隐式近似因式分解方法基本上可以描述非定常流动现象;(2)隐式亚迭代一阶时间精度格式会导致流场结构的变化,其精度可能是不足的;(3)隐式亚迭代二阶时间精度与简化Runge Kutta五步格式的计算结果一致,可以认为是计算非定常问题的适当方法.     

近尾迹流动的数值模拟

通过数值求解可压缩Navier-Stokes方程的方法来模拟近尾迹的分离流动。Navier-Stokes方程^[1]是利用中所给出的方法进行差分逼近的。这一方法兼备显式格式和隐式格式的优点。这是一隐式格式,因而可放松稳定性对时间步长的限制。这一格式又具有显式格式的简单性,差分方程的解可显式表达出来。求解过程中避免了通常隐式格式所要求的大量数值矩阵求逆和大量的矩阵运算。文中对不同的马赫数M_∞和雷诺数Re进行了计算。数值实验表明,本文所采用的方法是模拟底部分离流动的一个简单而有效的方法,可用来计算高马赫数和高雷诺数的分离流动。     

An adaptive implicit–explicit scheme for the DNS and LES of compressible flows on unstructured grids

An adaptive implicit–explicit scheme for Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) of compressible turbulent flows on unstructured grids is developed. The method uses a node-based finite-volume discretization with Summation-by-Parts (SBP) property, which, in conjunction with Simultaneous Approximation Terms (SAT) for imposing boundary conditions, leads to a linearly stable semi-discrete scheme. The solution is marched in time using an Implicit–Explicit Runge–Kutta (IMEX-RK) time-advancement scheme. A novel adaptive algorithm for splitting the system into implicit and explicit sets is developed. The method is validated using several canonical laminar and turbulent flows. Load balance for the new scheme is achieved by a dual-constraint, domain decomposition algorithm. The scalability and computational efficiency of the method is investigated, and memory savings compared with a fully implicit method is demonstrated. A notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed.     

A second order self-consistent IMEX method for radiation hydrodynamics

We present a second order self-consistent implicit/explicit (methods that use the combination of implicit and explicit discretizations are often referred to as IMEX (implicit/explicit) methods ,  and ) time integration technique for solving radiation hydrodynamics problems. The operators of the radiation hydrodynamics are splitted as such that the hydrodynamics equations are solved explicitly making use of the capability of well-understood explicit schemes. On the other hand, the radiation diffusion part is solved implicitly. The idea of the self-consistent IMEX method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations. In our self-consistent IMEX method, we solve the hydrodynamics equations inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-free Newton Krylov (JFNK) method ,  and . This is done to avoid order reductions in time convergence due to the operator splitting. We present results from several test calculations in order to validate the numerical order of our scheme. For each test, we have established second order time convergence.     

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes.     

A mixed implicit–explicit finite difference scheme for heat transport in magnetised plasmas

An explicit/implicit domain decomposition method is applied to the time-dependent heat-conduction problem in a 2-d, strongly anisotropic medium (a magnetised plasma), using a formulation of the spatial derivatives which avoids the pollution of perpendicular by parallel heat fluxes. The time-stepping at the sub-domain boundaries is done using a DuFort–Frankel scheme, which leads to a time step limit given not by instabilities, but by the damping rate of numerical oscillations driven by inconsistencies in the formulation of initial conditions or the temporal variations in the true physical solution. These limitations can be minimized, however, by aligning the subdomain boundaries as much as possible with magnetic flux surfaces. The time step limit depends on the ratio of the implicit grid spacing to the distance between subdomain boundaries (DuFort–Frankel lines in 2-d, surfaces in 3-d).     

具有TVD性质的三阶精度GODUNOV格式在粘性流场计算中的应用

本文发展了一种具有TVD性质的三阶精度的Godunov格式。隐式部分采用迎风对角线形隐式近似因式分解法。并引入了粘性通量的简化算法。显式部分采用三阶精度TVD格式。为进一步增强格式的稳定性及对间断的捕捉能力,在单元边界上构造Riemann问题。应用上述方法对某型涡轮高压级及压气机进行了数值模拟。     

Explicit TE/TM scheme for particle beam simulations

In this paper we propose an explicit two-level conservative scheme based on a TE/TM like splitting of the field components in time. Its dispersion properties are adjusted to accelerator problems. It is simpler and faster than the implicit version [I. Zagorodnov, T. Weiland, TE/TM scheme for computation of electromagnetic fields in accelerators, J. Comput. Phys. 207 (2005) 69]. It does not have dispersion in the longitudinal direction and the dispersion properties in the transverse plane are improved. The explicit character of the new scheme allows a uniformly stable conformal method without iterations and the scheme can be parallelized easily. It assures energy and charge conservation. A version of this explicit scheme for rotationally symmetric structures is free from the progressive time step reducing for higher order azimuthal modes as it takes place for Yee’s explicit method used in the most popular electrodynamics codes.     

一种新的LU型隐式格式及其应用

发展了一种新的ＬＵ型隐式格式。新格式除包含ＬＵ－ＳＧＳ格式全部特色外，尽管采用了相似变换而不是近似处理来精确地构造通量矢量的迎风Ｊａｃｏｂｉａｎ矩阵，仍可避免块对角矩阵求逆．且左端项仍可保持矢量化处理。新格式显示出更快的收敛性和更高的稳定性且没有多余的数值耗散和自由参数，尤其适用于求解三维非定常流动问题。     

Development of an explicit algorithm for remote sensing estimation of chlorophyll a using symbolic regression

The primary mission of ocean color remote sensing is to provide accurate marine bio-optical properties from satellite data. We propose a new algorithm that uses symbolic regression to estimate chlorophyll a (chl a) concentrations from remote sensing reflectance. We compared the accuracy and computational efficiency of the new algorithm to that of the explicit empirical algorithms (OC4v4 and OC4v6), and implicit algorithms based on neural networks or support vector machines (SVM). Results show that the accuracy of the symbolic regression algorithm is higher than that of the OC4 algorithms and comparable to that of implicit algorithms. The improvement is particularly important for high biomass areas (chl a?≧?3 mg?m^-3) that are often found in optically complex waters. The computational efficiency of the explicit algorithm developed by symbolic regression is comparable to that of the two versions of OC4 algorithms and better than that of implicit algorithms based on SVM. With its good precision and fast processing, the symbolic regression algorithm is a powerful tool for remote sensing of chl a that could be used advantageously in the reprocessing of large datasets.     

Implicit subgrid-scale modeling by adaptive deconvolution

A new approach for the construction of implicit subgrid-scale models for large-eddy simulation based on adaptive local deconvolution is proposed. An approximation of the unfiltered solution is obtained from a quasi-linear combination of local interpolation polynomials. The physical flux function is modeled by a suitable numerical flux function. The effective subgrid-scale model can be determined by a modified-differential equation analysis. Discretization parameters which determine the behavior of the implicit model in regions of developed turbulence can be adjusted so that a given explicit subgrid-scale model is recovered to leading order in filter width. Alternatively, improved discretization parameters can be found directly by evolutionary optimization. Computational results for stochastically forced and decaying Burgers turbulence are provided. An assessment of the computational experiments shows that results for a given explicit subgrid-scale model can be matched by computations with an implicit representation. A considerable improvement can be achieved if instead of the parameters matching an explicit model discretization parameters determined by evolutionary optimization are used.     

平面光波导器件时域有限差分束传输法分析中边界和初始条件的处理

从慢变包络（SVEA）时域光传输方程出发,对比了综合道格拉斯（GD）和传统的中心差分（CN）两种离散化格式,指出GD法比CN法精度有了很大提高.重点讨论了GD法的TD-BPM边界处理问题,给出了吸收边界和透明边界离散的显式和隐式表达式,分析并确定了交替方向隐式法（ADIM）的边界处理方式;同时,对激励源加入的连接边界进行了仔细讨论.结果表明对于隐式GD-TD-BPM法边界处理应采用隐式,中间激励更能反映光波导反射场的情况.     

对流方程的GE方法

用显式迎风格式或隐式迎风格式给出了求解对流方程的GE方法,证明了该方法的相容性及(弱)稳定性,数值例子说明方法是良好的.     

Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier–Stokes equations

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier–Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier–Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge–Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5–10 compared with a well tuned E-RK scheme.     

粒子输运离散纵标方程基于界面修正的并行计算方法

为了改造粒子输运方程求解的隐式格式,研究设计适应大型并行计算机的并行计算方法,介绍一类求解粒子输运方程离散纵标方程组的基于界面修正的源迭代并行计算方法.应用空间区域分解,在子区域内界面处首先采用迎风显式差分格式进行预估,构造子区域的入射边界条件,然后,在各个子区域内部进行源迭代求解隐式离散纵标方程组.在源迭代过程中,在内界面入射边界处采用隐式格式进行界面修正.数值算例表明该并行计算方法在精度、并行度、简单性诸方面均具有良好的性质.     

Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows

The localized artificial diffusivity method is investigated in the context of large-eddy simulation of compressible turbulent flows. The performance of different artificial bulk viscosity models are evaluated through detailed results from the evolution of decaying compressible isotropic turbulence with eddy shocklets and supersonic turbulent boundary layer. Effects of subgrid-scale (SGS) models and implicit time-integration scheme/time-step size are also investigated within the framework of the numerical scheme used. The use of a shock sensor along with artificial bulk viscosity significantly improves the scheme for simulating turbulent flows involving shocks while retaining the shock-capturing capability. The proposed combination of Ducros-type sensor with a negative dilatation sensor removes unnecessary bulk viscosity within expansion and weakly compressible turbulence regions without shocks and allows it to localize near the shocks. It also eliminates the need for a wall-damping function for the bulk viscosity while simulating wall-bounded turbulent flows. For the numerical schemes used, better results are obtained without adding an explicit SGS model than with SGS model at moderate Reynolds number. Inclusion of a SGS model in addition to the low-pass filtering and artificial bulk viscosity results in additional damping of the resolved turbulence. However, investigations at higher Reynolds numbers suggest the need for an explicit SGS model. The flow statistics obtained using the second-order implicit time-integration scheme with three sub-iterations closely agrees with the explicit scheme if the maximum Courant–Friedrichs–Lewy is kept near unity.     

The Darwin Direct Implicit Particle-in-Cell (DADIPIC) Method for Simulation of Low Frequency Plasma Phenomena

We describe a new algorithm for simulating low frequency, kinetic phenomena in plasmas. Darwin direct implicit particle-in-cell (DADIPIC), as its name implies, is a combination of the Darwin and direct implicit methods. Through the Darwin method the hyperbolic Maxwell's equations are reformulated into a set of elliptic equations. Propagating light waves do not exist in the formulation so the Courant constraint on the time step is eliminated. The direct implicit method is applied only to the electrostatic field with the result that electrostatic plasma oscillations do not have to be resolved for stability. With the elimination of these constraints spatial and temporal discretization can be much larger than that possible with explicit, electrodynamic PIC. We discuss the algorithms for pushing the particles and solving for the fields in 2D cartesian geometry. We also detail boundary conditions for conductors and dielectrics. Finally, we present two test cases, electron cyclotron waves and collisionless heating in inductively coupled plasmas. For these test cases DADIPIC shows agreement with analytic kinetic theory and good energy conservation characteristics.     

Explicit and implicit finite difference schemes for fractional Cattaneo equation

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor–corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor–corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.     

Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations

We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.     

MLIMCL：基于机器学习的隐式溶剂蒙特卡罗方法

对于分子结构的优化和预测,蒙特卡罗(MC)是很重要的计算工具. 当溶剂效应被显式的考虑时,由于水分子和电离子的自由度很大,蒙特卡罗方法变得非常昂贵. 相对而言,基于隐式溶剂的蒙特卡罗方法可以通过对溶剂效应平均场的近似来大大降低计算成本,同时还能保持目标分子在原子水平上的细节. 目前两种最流行的隐式溶剂模型是泊松-波兹曼模型和通用化波恩模型. 通用化波恩模型是泊松-波兹曼模型的近似,但在模拟计算时间上要快得多. 本文通过结合两种隐式溶剂模型在准确性和效率方面的优势,开发了一种基于机器学习的隐式溶剂蒙特卡罗方法. 具体而言,蒙特卡罗方法通过机器学习既保留了泊松-波兹曼模型的精度,又达到了通用化波恩模型的速度,从而能快速准确地获取模拟计算中每一步的静电溶解自由能. 本文采用苯-水系统和蛋白质-水系统来验证我们的蒙特卡罗方法. 实验证明蒙特卡罗方法在分子结构优化和预测的速度和准确性方面具有很大优势.