任意阶显式精细积分多步法的常用形式及其高阶次数值计算

基于任意阶显式精细积分多步法的一般公式,给出其几种常用形式,并实现了高阶次数值计算,将新算法应用于射线方程和双原子系统经典轨迹数值计算中.数值计算结果表明任意阶显式精细积分多步法是一种高精度、高效率、稳定性较好的方法,并且可方便地进行高阶次的运算.     

非线性系统的任意项精细积分外插多步法及其在混沌数值分析中的应用

对非线性系统提出了高精度的精细积分任意项外插多步法的计算公式.本方法只需增加插值 项数即可提高计算精度，同时不会增加过大的计算量，发展完善了精细积分法.将本方法应 用于混沌方程中，取得了较好的效果.数值计算结果表明，该方法是一种高精度、高效率的 方法，在求解混沌系统上比传统方法有很大的优势. 关键词： 非线性系统 任意项精细积分多步法 混沌系统     

空间分数阶导数“反常”扩散方程数值算法的比较

分别采用显式差分格式、隐式差分格式以及Crank-Nicholson差分格式数值求解空间分数阶导数,并从局部截断误差、稳定性、计算量三个方面进行比较分析;通过数值算例验证分析结果.     

二维单步交替方向隐式时域有限差分法吸收边界性能分析

给出了两种适用于二维单步交替方向隐式时域有限差分(2-D Leapfrog ADI-FDTD)方法的吸收边界:Mur边界和卷积完全匹配层(CPML)边界。单步交替方向隐式时域有限差分(Leapfrog ADI-FDTD)方法是一种无条件稳定的全隐式差分算法,由于二维空间Leapfrog ADI-FDTD的迭代同时存在显式和隐式方程,故而不同电磁分量的边界条件也存在差异。从原理出发,推导了适用于2-D Leapfrog ADI-FDTD方法的CPML边界条件,并与一阶Mur边界进行比较,利用自由空间的反射误差来表征两种边界的吸收性能,简要总结了两种吸收边界的优缺点。     

一类n阶三脚架电阻网络的等效电阻研究

研究了一类n阶三脚架网络的等效电阻模型,该模型含有7个不同的电阻元素,因而包含了多个网络模型.文章采用构建等效模型的方法导出了一个非线性差分方程模型,采用变量代换的方法间接地给出了非线性差分方程的通解.本文进一步创造了一个负电阻的概念,获得了电阻网络任意节点间的等效电阻公式.利用特殊条件下的数个特殊例子与相关结果进行了比较.本文的结论也适用于复阻抗网络的等效复阻抗研究.     

任意马赫数非定常流动数值模拟的统一算法

发展适用于从低速到高速任意马赫数非定常流动数值模拟的统一算法.通过引入一个伪时间导数项和一个新的预处理矩阵,得到双时间非定常预处理可压缩Navier-Stokes方程.方程的对流项采用三阶Roe通量近似差分格式离散,粘性项采用二阶中心差分格式离散.基于数值通量的线性化技术,实现伪时间步的隐式ADI-LU格式迭代,进而获得物理时间步的二阶推进精度.重点以低马赫数流动为例,求解了圆柱绕流和NACA0015翼型等速上仰动态失速问题.计算结果表明该统一算法能够较好地模拟低马赫数乃至任意马赫数非定常流动.     

基于差分进化算法的核磁共振T2谱多指数反演

提出了一种基于差分进化(DE)算法的核磁共振弛豫信号多指数反演新方法. 将核磁共振T₂谱反演问题转化为带非负约束的非线性优化问题,不需要预先给定横向弛豫时间T_{2分布,直接利用差分进化算法进行反演计算. 在测量信号低信噪比的情况下,计算机模拟和实验数据反演都表明了该方法在分析处理NMR弛豫信号中的有效性. 关键词： 核磁共振 多指数反演 差分进化 岩心分析}     

一种新时域交替隐式差分算法在散射问题中的应用

提出一种新时域交替隐式有限差分(ADI-FDTD)算法格式. 传统ADI-FDTD算法的 场量步进方程涉及周围若干网格的较多场量，导致两个区域的步进方程处理较困难：一个是 邻近完全匹配层(PML)和散射场交界区，另一个是邻近连接边界区. 特别是后者，考虑入射 波影响需对场量所在区域判断，根据不同情况对原有方程进行修正，一维和二维散射问题 相对简单，可三维问题修正式有数十种之多而几乎无法完成. 本方法基于分裂场形式的ADI- FDTD技术，使得散射场区和PML吸收层区的表达形式完全一致，从而忽略两者差别.另 关键词： 时域交替隐式有限差分算法 电磁散射     

非线性热传导方程的分组隐式解法及数值结果

对于激光物理研究中提出的一个非线性热传导问题,利用Saul'yev非对称格式构造了适合并行计算的分组隐式解法.数值试验表明,新算法与通常的隐式算法计算精度相当,迭代次数相同.     

谐振管内非线性驻波的有限体积数值算法

为了扩展谐振管内非线性驻波在工程中的应用, 以及克服现有数值计算方法仅局限于求解直圆柱形和指数形谐振管内非线性驻波的问题. 根据变截面的非稳态可压缩热黏性流体Navier-Stokes方程和空间守恒方程, 并基于求解压力速度耦合方程的半隐式算法和交错网格技术, 构建一种能够计算任意形状轴对称谐振管受活塞驱动时内部非线性驻波的有限体积算法. 分别对圆柱形、指数形和圆锥形谐振管内的非线性驻波进行仿真计算. 通过与现有试验结果以及数值仿真结果的对比, 验证了该方法的正确性.并获得除驻波声压之外的另外一些新的物理结果, 包括速度、密度、温度的瞬时变化.在直圆柱形谐振管内产生冲击声压波, 速度波形中出现钉状结构.而在指数形和圆锥形谐振管内产生高声压幅值的驻波, 没有出现冲击波, 速度波形中均未发现钉状结构. 计算结果表明谐振管内非线性驻波的物理属性与谐振管形状之间有密切关系.     

New Predictor-Corrector Methods Based on Exponential Time Differencing for Systems of Nonlinear Differential Equations

We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams-Bashforth-Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.     

An implementation of the exponential time differencing scheme to the magnetohydrodynamic equations in a spherical shell

Over the last decade there has been renewed interest in applying exponential time differencing (ETD) time stepping schemes to the solution of stiff systems. In this paper, we present an implementation of such a scheme to the fully spectral solution of the incompressible magnetohydrodynamic equations in a spherical shell. One problem associated with ETD schemes is the accurate calculation of the necessary matrices; we implement and discuss in detail a variety of different methods including direct computation, contour integration, spectral expansions and recurrence relations. We compare the accuracy of six different second-order methods in determining the evolution of a three-dimensional magnetic field under the action of a prescribed time-dependent flow of electrically conducting fluid, and find that for the timestep restriction imposed by the nonlinear terms, ETD methods are no more accurate than linearly implicit methods which have the significant advantage of being easier to implement. However, ETD methods are more readily extendable than those which are linearly implicit and will become much more advantageous at higher order.     

Finite-Difference Time-Domain Algorithm for Dispersive Media Based on Runge-Kutta Exponential Time Differencing Method

The electromagnetic propagation in dispersive media is modeled using finite difference time domain (FDTD) method based on the Runge-Kutta exponential time differencing (RKETD) method. The second-order RKETD-FDTD formulation is derived. The high accuracy and efficiency of the presented method is confirmed by computing the transmission and reflection coefficients for a nonmagnetized collision plasma slab in one dimension. The comparison of the numerical results of the RKETD and the exponential time differencing (ETD) algorithm with analytic values indicates that the RKETD is more accurate than the ETD algorithm.     

Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods

This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable.     

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.     

Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods

Integration factor methods are a class of “exactly linear part” time discretization methods. In [Q. Nie, Y.-T. Zhang, R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006) 521–537], a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. In this paper, we solve this problem by applying the Krylov subspace approximations to the matrix exponential operator. Then we apply this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction–diffusion equations. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the method in resolving the stiffness of the DG spatial operator for reaction–diffusion PDEs. Application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown.     

Nonlinear fluctuation,frequency and stability analyses in free vibration of circular sector oscillation systems

In the present paper, the modified homotopy perturbation method (MHPM) is employed to investigate about both nonlinear swinging oscillation and the stability of circular sector oscillation systems. The sensitivity study performed for frequency analysis of the mentioned oscillatory circular sector body shows that frequency of nonlinear oscillation depends on some specific parameters and can be optimized. Furthermore onset of the instability is dependent to angle α and initial amplitude.Comparisons made among the results of the present closed-form analytical solution and the traditional numerical iterative time integration solution confirms the accuracy and efficiency of the presented analytical solution.In contrast to the available numerical methods, the present analytical method is free from the numerical damping and the time integration accumulated errors. Moreover, in comparison with the traditional multistep numerical iterative time integration methods, a much less computational time is required for the present analytical method. Responses of the dynamical systems to some extent are affected by the natural frequencies. Results reveal that for nonlinear systems, the natural frequency is remarkably affected by the initial conditions.     

Shift operator method and Runge-Kutta exponential time differencing method for plasma

In this paper, the shift operator finite difference time domain (SO-FDTD) method and Runge-Kutta exponential time differencing (RKETD) method are introduced. The high accuracy and efficiency of the two methods are verified by calculating the reflection and transmission coefficients of electromagnetic waves through a collisional plasma slab. A comparison of computational efficiency of the two methods is presented by simulating the electromagnetic wave propagation in homogeneous non-magnetized plasma. The numerical results indicate that the calculation time using SO-FDTD method is less than that using RKETD method with almost the same accuracy.     

Exponential Time Differencing Algorithm for Debye Medium in FDTD

The exponential time differencing (ETD) formultion is proposed for Debye medium using finite-difference time-domain (FDTD) method. The schemes of the auxiliary differential equation (ADE) and ETD algorithms are presented. The algorithms are validated by comparing the simulation results with analytical values. The ETD method costs the same memory as the ADE method, and the numerical results indicate that the ETD technique is more accurate than the ADE method. This work is supported by the foundation under Grant No. Zl 0502.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.