一维对流扩散反应方程的高精度数值方法

通过将原方程变换为对流扩散方程,将所得方程的对流项采用四阶组合紧致迎风格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的空间半离散格式采用四阶龙格库塔方法进行时间推进,得到了一种求解非定常对流扩散反应问题的高精度方法,其收敛阶为O(h4+τ4).经数值实验并与文献结果进行对比,表明该格式适用于对流占优问题的数值模拟,验证了格式的良好性能.     

Riesz回火分数阶平流-扩散方程的隐式中点方法

本文针对Riesz回火分数阶平流-扩散方程,采用隐式中点方法离散一阶时间偏导数,用修正的二阶Lubich回火差分算子逼近Riesz空间回火分数阶偏导数,并对平流项采用中心差商进行离散,构造出新的数值方法,获得了数值方法的稳定性和收敛性,该方法的收敛阶在空间和时间方向均达到二阶精度.数值试验验证了数值方法的有效性.     

离散奇异内积法分析材料非线性柱的动力响应

引入离散奇异内积法分析材料非线性圆柱的动力响应．离散奇异内积方法是一种结合全局方法的高精度和局域方法的稳定性的计算方法．数值分析过程中用离散奇异内积方法离散空间导数,用四阶Runge—Kutta法离散时间导数．计算结果表明,离散奇异内积格式的求解结果和LP法的求解结果非常吻合．说明离散奇异内积格式非常适合数值分析材料非线性圆柱的动力响应问题,并且是一种具有很高的精度,和可靠性的高效的算法。     

四阶高分辨率熵相容算法

针对一维Burgers方程和一维Euler方程组的数值求解问题,提出了一种四阶高分辨率熵相容算法。新算法时间方向采用半离散方式,空间方向应用四阶中心加权基本无振荡(CWENO)重构方法,数值通量引入Ismail通量函数,将新的四阶算法应用于静态激波问题、激波管问题以及强稀疏波问题的数值求解中,并将所得结果同准确解以及已有算法所得结果进行了分析与比较。数值结果表明:新算法计算结果正确、分辨率高,能够准确捕捉激波及稀疏波,并能有效避免膨胀激波的产生。新算法适用于准确解决一维Burgers方程和一维Euler方程组的数值求解问题。     

一维非定常对流扩散方程的高阶组合紧致迎风格式

通过将对流项采用四五阶组合迎风紧致格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的半离散格式在时间方向采用四阶龙格库塔方法求解,从而得到了一种求解非定常对流扩散方程问题的高精度组合紧致有限差分格式,其收敛阶为O(h~4+τ~4).经Fourier精度分析和数值验证,证实了格式的良好性能.三个数值算例包括线性常系数问题,矩形波问题和非线性问题,数值结果表明:该格式具有很高的分辨率,且适用于对高雷诺数问题的数值模拟.     

求解复杂交通流模型的低耗散中心迎风格式

针对非均匀道路上的多车种LWR交通流模型,提出一种低耗散中心迎风格式。以4阶中心加权基本无震荡重构和低耗散中心迎风数值通量为基础,通过构造不同形式的全局光滑因子及增大非光滑模板对应的非线性权重优化数值格式的耗散特性,并采用Runge-Kutta方法对半离散数值格式在时间方向上进行离散使其保持4阶精度。对非均匀道路上多车种LWR交通流模型的车道数变化和交通信号灯控制问题进行数值模拟,结果表明该格式具有4阶求解精度,且分辨率高。     

连续时间 Hopfield网络模型数值实现分析

讨论使用Euler方法和梯形方法在数值求解连续时间的Hopfield网络模型时,离散时间步长的选择和迭代停止条件问题.利用凸函数的定义研究了能量函数下降的条件,根据凸函数的性质分析它的共轭函数减去二次函数之差仍为凸函数的条件.分析连续时间Hopfield网络模型的收敛性证明,提出了一个广义的连续时间Hopfield网络模型.对于常用的Euler方法和梯形方法数值求数值实现连续时间Hopfield网络,讨论了离散时间步长的选择.由于梯形方法为隐式方法,分析了它的迭代求算法的停止条件.根据连续时间Hopfield网络的特点,提出改进的迭代算法,并对其进行了分析.数值实验的结果表明,较大的离散时间步长不仅加速了数值实现,而且有利于提高优化性能.     

粘性Burgers方程的高阶隐式WCNS格式

JFNK (Jacobian-free Newton-Krylov)方法是由外层Newton迭代法和内层Krylov子空间迭代法构成的嵌套迭代方法.本文提出了一种基于JFNK方法的高阶隐式WCNS (weighted compact nonlinear scheme)格式,并用于求解一维、二维粘性Burgers方程.外层迭代法采用含参数的多步Newton迭代法,给出了收敛性分析,内层迭代法采用无矩阵GMRES迭代法.粘性Burgers方程的非线性对流项采用五阶WCNS格式计算.为提高方法精度和计算效率,时间离散采用三阶隐式的DIRK (diagonal implicit Runge-Kutta)方法.数值结果表明基于JFNK方法的隐式WCNS格式在时间上能达到三阶精度,与显式TVD Runge-Kutta WCNS方法相比,计算效率更高.此外,基于JFNK方法的隐式WCNS格式稳定性好,且具有良好的激波捕捉能力.     

一维非线性系统FPK 方程的TVD Runge⁃Kutta WENO 型差分解

首先研究了非线性随机动力系统所对应的Fokker-Planck-Kolmogorov(FPK)方程.其次,讨论了微分方程的三阶TVD Runge-Kutta关于时间的离散差分格式以及关于空间离散的五阶Weighted Essentially nonOscillatory(WENO)差分格式,并将其相结合,得到FPK方程的TVD Runge-Kutta WENO差分解,并与FPK方程的精确解进行了比较.数值结果表明,该方法具有良好的稳定性,且可以解决其他方法在概率密度峰值处偏小,而在尾部处较大等缺点.     

Wright分数阶时滞微分方程的离散化过程

引进了一种离散化方法对分数阶时滞微分方程进行离散化求解。首先考察Wright分数阶时滞微分方程;其次分析相应具有分段常数变元的Wright分数阶时滞微分方程,并应用离散化过程对模型进行数值求解;然后根据不动点理论讨论该合成动力系统不动点的稳定性;最后借助MATLAB对模型进行数值仿真,并结合Lyapunov指数、相图、时间序列图、分岔图探讨模型更多复杂的动力学现象。结果显示,提出方法成功对Wright分数阶时滞微分方程进行离散。     

Four-stage approximation in time-dependent ionization processes

A limited storage method for treating time-dependent ionization processes in plasma simulations is described. The method is implicit so that arbitrary timesteps may be taken. For large timesteps the solution approaches the equilibrium solution. Variations of the method are presented with numerical examples.     

An accurate interest matching algorithm based on prediction of the space–time intersection of regions for the distributed virtual environment

Interest matching is an important data-filtering mechanism for a large-scale distributed virtual environment. Many of the existing algorithms perform interest matching at discrete timesteps. Thus, they may suffer the missing-event problem: failing to report the events between two consecutive timesteps. Some algorithms solve this problem, by setting short timesteps, but they have a low computing efficiency. Additionally, these algorithms cannot capture all events, and some spurious events may also be reported. In this paper, we present an accurate interest matching algorithm called the predictive interest matching algorithm, which is able to capture the missing events between discrete timesteps. The PIM algorithm exploits the polynomial functions to model the movements of virtual entities, and predict the time intervals of region overlaps associated with the entities accurately. Based on the prediction of the space–time intersection of regions, our algorithm can capture all missing events and does not report the spurious events at the same time. To improve the runtime performance, a technique called region pruning is proposed and used in our algorithm. In experiments, we compare the new algorithm with the frequent interest matching algorithm and the space–time interest matching algorithm on the HLA/RTI distributed infrastructure. The results prove that although an additional matching effort is required in the new algorithm, it outperforms the baselines in terms of event-capturing ability, redundant matching avoidance, runtime efficiency and scalability.     

An adaptive finite element scheme for transient problems in CFD

An adaptive finite element scheme for transient problems is presented. The classic h-enrichment / coarsening is employed in conjunction with a triangular finite element discretization in two dimensions. A mesh change is performed every n timesteps, depending on the Courant number employed and the number of ‘protective layers’ added ahead of the refined region. In order to simplify the refinement/ coarsening logic and to be as fast as possible, only one level of refinement/coarsening is allowed per mesh change. A high degree of vectorizability has been achieved on the CRAY XMP 12 at NRL. Several examples involving shock-shock interactions and the impact of shocks on structures demonstrate the performance of the method, indicating that considerable savings in CPU time and storage can be realized even for strongly unsteady flows.     

Multirate Timestepping Methods for Hyperbolic Conservation Laws

This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps (restricted by the largest value of the Courant number on the grid) and therefore results in more efficient algorithms. Numerical results obtained for the advection and Burgers’ equations confirm the theoretical findings. This work was supported by the National Science Foundation through award NSF CCF-0515170.     

Radiative cooling in numerical astrophysics: The need for adaptive mesh refinement

Energy loss through optically thin radiative cooling plays an important part in the evolution of astrophysical gas dynamics and should therefore be considered a necessary element in any numerical simulation. Although the addition of this physical process to the equations of hydrodynamics is straightforward, it does create numerical challenges that have to be overcome in order to ensure the physical correctness of the simulation. First, the cooling has to be treated (semi-)implicitly, owing to the discrepancies between the cooling timescale and the typical timesteps of the simulation. Secondly, because of its dependence on a tabulated cooling curve, the introduction of radiative cooling creates the necessity for an interpolation scheme. In particular, we will argue that the addition of radiative cooling to a numerical simulation creates the need for extremely high resolution, which can only be fully met through the use of adaptive mesh refinement.     

Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations

Implicit methods for finite-volume schemes on unstructured grids typically rely on a matrix-free implementation of GMRES and an explicit first-order accurate Jacobian for preconditioning. Globalization is typically achieved using a global timestep or a CFL based local timestep. We show that robustness of the globalization can be improved by supplementing the pseudo-timestepping method commonly used with a line search method. The number of timesteps required for convergence can be reduced by using a timestep that scales with the local residual. We also show that it is possible to form the high-order Jacobian explicitly at a reasonable computational cost. This is demonstrated for cases using both limited and unlimited reconstruction. This exact Jacobian can be used for preconditioning and directly in the GMRES method. The benefits of improvements in preconditioning and the elimination of residual evaluations in the inner iterations of the matrix-free GMRES method are substantial. Computational results focus on second- and fourth-order accurate schemes with some results for the third-order scheme. Overall computational cost for the matrix-explicit method is lower than the matrix-free method for all cases. The fourth-order matrix-explicit scheme is a factor of 1.6-3 faster than the matrix-free scheme while requiring about 50-100% more memory.     

基于多变量密码体制的签密方案

为了解决基于传统公钥密码的签密方案不能抵抗量子攻击的问题,提出了一种基于多变量公钥密码的签密方案。结合多层Matsumoto-Imai(MMI)方案中心映射的多层构造、CyclicRainbow签名方案,以及隐藏域方程(HFE)的中心映射构造,提出了一种改进的中心映射构造方法,并由此设计了相应的签密方案。分析表明,所设计的方案与MMI方案相比,在实现了加密和签名的同时,方案密钥量和密文量分别减少了5%和50%。在随机预言模型下,基于多变量方程组求解困难问题假设和多项式同构困难问题假设,证明了该方案在适应性选择密文攻击下具有不可区分性,在适应性选择消息攻击下具有不可伪造性。     

基于物理模型的水波动画

近年来,自然景物的模拟一直是计算机图形学中最具挑战性的问题之一。关于山、水、云、烟、火焰等自然景物的模拟,在计算机游戏、影视、广告等各种领域中有着广泛的用途。作为自然景物模拟的重要内容,对流水、波浪的模拟正日益引起人们的关注。本文基于二维浅水波方程模型,给出了一种实现水波动画的数值模拟方法。和以往的方法不同的是,本文既没有去特意构造具体的波形函数,也没有去求解复杂的Navier-Stokes方程,而是基于二维浅水波方程模型,采用隐式半拉格朗日积分方法进行求解,在保证稳定性的同时,可以允许大时间步长。实践证明,用该算法模拟水波,效果比较真实,而且在普通的PC平台上即可满足一般动画的实时需求。     

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.     

A high-order non-oscillatory central scheme with non-staggered grids for hyperbolic conservation laws

In this work, we present a scheme which is based on non-staggered grids. This scheme is a new family of non-staggered central schemes for hyperbolic conservation laws. Motivation of this work is a staggered central scheme recently introduced by A.A.I. Peer et al. [A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws, Appl. Numer. Math. 58 (2008) 674–688]. The most important properties of the technique developed in the current paper are simplicity, high-resolution and avoiding the use of staggered grids and hence is simpler to implement in frameworks which involve complex geometries and boundary conditions. Numerical implementation of the new scheme is carried out on the scalar conservation laws with linear, non-linear flux and systems of hyperbolic conservation laws. The numerical results confirm the expected accuracy and high-resolution properties of the scheme.