时间分数阶反应-扩散方程混合差分格式的并行计算方法

分数阶反应-扩散方程有深刻的物理和工程背景,其数值方法的研究具有重要的科学意义和应用价值.文中提出时间分数阶反应-扩散方程混合差分格式的并行计算方法,构造了一类交替分段显-隐格式(alternative segment explicit-implicit,ASE-I)和交替分段隐-显格式(alternative segment implicit-explicit,ASI-E),这类并行差分格式是基于Saul'yev非对称格式与古典显式差分格式和古典隐式差分格式的有效组合.理论分析格式解的存在唯一性,无条件稳定性和收敛性.数值试验验证了理论分析,表明ASE-I格式和ASI-E格式具有理想的计算精度和明显的并行计算性质,证实了这类并行差分方法求解时间分数阶反应-扩散方程是有效的.     

双项时间分数阶慢扩散方程的一类高效差分方法

反常扩散既是一个重要的物理课题,也是工程中普遍涉及的一个现实问题.针对双项时间分数阶慢扩散方程,本文结合古典显式格式和古典隐式格式,提出了显-隐(Explicit-Implicit,E-I)差分方法和隐-显(Implicit-Explicit,I-E)差分方法.分析证明E-I格式解和I-E格式解的存在唯一性,稳定性和收敛性.理论分析和数值试验结果均表明E-I和I-E差分方法无条件稳定,具有空间2阶精度、时间2-α阶精度.在计算精度一致的要求下,E-I和I-E差分方法相较于经典隐式差分方法具有省时性,证实了E-I差分方法和I-E差分方法求解双项时间分数阶慢扩散方程是高效可行的.     

铁磁链方程的Fourier谱方法和拟谱方法

在铁磁链方程运动研究中，各项同性Heisellberg链的所谓Landau－Lifshitz方程L‘1为其中旋密度Z＝（。，t），w）”和h＝（0，0，h（t））”为三维向量函数，。X”表示三维向量的叉积．这种方程组还常在凝聚态介质物理的问题中出现，有不少文章是关于Landau－Lifshitz方程组的孤立于解，孤立波的相互作用以及无穷守恒律等的研究［‘－‘］，［5，6，7］研究了具有小扩散项旋方程组解的存在性及隐式差分格式．在［7］中给出的结果，证明了铁磁连方程解的存在性与唯一性，作者在［8］中考察了旋方程组（2）的周期初值问题的显式差分解，并…     

时间分数阶慢扩散方程的一类有效差分方法

对时间分数阶慢扩散方程提出一类数值差分方法:显-隐(Explicit-Implicit, E-I)和隐-显(Implicit-Explicit, I-E)差分方法.它是将古典显式格式与古典隐式格式相结合构造出的一类有效差分格式.理论证明了格式解的存在唯一性,用傅里叶方法证明了格式的稳定性和收敛性.数值试验验证了理论分析,表明E-I格式和I-E格式在具有良好的精度且无条件稳定的情况下,计算速度比隐式格式提高了75%.从而用此格式解决分数阶慢扩散方程是可行的.     

解非线性二阶双曲型方程的一种新数值方法

本文建立了解二阶双曲型方程的一种新数值方法一再生核函数法．利用再生核函数，直接给出每个离散时间层上近似解的显式表达式．此方法的优点是：计算格式绝对稳定，且可显式求解；利用显式表达式，可实现完全并行计算等文中对近似解的收敛性和稳定性进行了理论分析，并给出数值算例．     

高阶抛物型方程的具有高稳定性的显式与半显式差分格式

高阶抛物型方程的具有高稳定性的显式与半显式差分格式曾文平（华侨大学数学系，泉州３６２０１１）１引言１９６０年Ｃａｖｅｂ在文［１］中，讨论了如下的高阶抛物型方程混合问题提出了一类含权因子α（０≤α≤１）的两层差分格式（初边值条件处理同［１］下同，从略）...     

广义非线性Sine-Gordon方程的两个隐式差分格式

本文对一类非线性Sine-Gordon方程的初边值问题提出了两个隐式差分格式．两个隐式差分格式的精度均为O(τ~2 h~2)．我们用离散泛函分析的方法证明了格式的收敛性和稳定性,并证明了求解格式的追赶迭代法的收敛性,最后给出了数值结果．结果表明本文的格式是有效的和可靠的．     

常微分方程的半隐Runge-Kutta法

针对某些非线性常微分方程,提出一种算子分裂半隐Runge-Kutta方法,对于非线性部分采用显式计算,对于刚性强的线性部分采用隐式处理.给出了格式的推导,分析了绝对稳定性,并证明了半隐二阶格式的收敛性.相比于显式Runge-Kutta法,半隐格式计算量相近,但改进了稳定性,数值结果显示了方法的合理性和有效性.最后,将算子分裂半隐Runge-Kutta方法应用于数值求解Zakharov偏微分方程组.     

三阶非线性KdV方程的交替分段显-隐差分格式

对三阶非线性KdV方程给出了一组非对称的差分公式,用这些差分公式与显、隐差分公式组合,构造了一类具有本性并行的交替分段显-隐格式A·D2证明了格式的线性绝对稳定性．对1个孤立波解、2个孤立波解的情况分别进行了数值试验．数值结果显示,交替分段显-隐格式稳定,有较高的精确度．     

求解隐式差分方程的加速迭代并行算法

1、引言 近年来,求解抛物型方程的有限差分并行迭代算法有了较大发展．针对稳定性好且难于并行化的隐式差分方程,文第一次提出了构造分段隐式的思想,建立了分段显-隐式（ASE-Ⅰ）方法和交替分段Crank-Nicolson（ASC-N）方法,实现了分而治之原则,     

Approximate inversion of the 3 D radon transform

The nuclear magnetic resonance (NMR) zeugmatography, a new technology in diagnostic medicine, leads to the problem of inverting the Radon transform in three dimensions. In contrast to even dimensions in odd dimensions the inversion formula is local. In three dimensions it consists of a backprojection operator, which brings no problems for the implementation, and taking the second derivative of the data. This is usually approximated by the central difference quotient. Here the effect of this approximate inversion operator is studied, error estimates for this ill-posed problem are given and a stepsize is computed which minimizes the sum of discretization and data error.     

A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives

In this study a new framework for solving three-dimensional (3D) time fractional diffusion equation with variable-order derivatives is presented. Firstly, a θ-weighted finite difference scheme with second-order accuracy is introduced to perform temporal discretization. Then a meshless generalized finite difference (GFD) scheme is employed for the solutions of remaining problems in the space domain. The proposed scheme is truly meshless and can be used to solve problems defined on an arbitrary domain in three dimensions. Preliminary numerical examples illustrate that the new method proposed here is accurate and efficient for time fractional diffusion equation in three dimensions, particularly when high accuracy is desired.     

Modeling shape distributions and inferences for assessing differences in shapes

The general class of complex elliptical shape distributions on a complex sphere provides a natural framework for modeling shapes in two dimensions. Such class includes many distributions, e.g., complex Normal, Watson, Bingham, angular central Gaussian and several others. We employ this class of distributions to develop methods for asserting differences in populations of shapes in two dimensions. Maximum likelihood and Bayesian methods for estimation of modal difference are developed along with hypothesis testing and credible regions for average shape difference. The methodology is applied in an example from biometry, where we are interested in detecting shape differences between male and female gorilla skulls.     

构造二维双曲型方程完全守恒差分格式的一种方法

§1 许多物理过程(例如气动力学,激光等离子体相互作用,磁流体力学,基本粒子输运等)的数学模型均可写成偏导数形式的二维不定常偏微分方程组:     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.     

The optimal relaxation parameter for the SOR method applied to the Poisson equation in any space dimensions

The finite difference discretization of the Poisson equation with Dirichlet boundary conditions leads to a large, sparse system of linear equations for the solution values at the interior mesh points. This problem is a popular and useful model problem for performance comparisons of iterative methods for the solution of linear systems. To use the successive overrelaxation (SOR) method in these comparisons, a formula for the optimal value of its relaxation parameter is needed. In standard texts, this value is only available for the case of two space dimensions, even though the model problem is also instructive in higher dimensions. This note extends the derivation of the optimal relaxation parameter to any space dimension and confirms its validity by means of test calculations in three dimensions.     

An efficient high‐order algorithm for solving systems of reaction‐diffusion equations

An efficient higher‐order finite difference algorithm is presented in this article for solving systems of two‐dimensional reaction‐diffusion equations with nonlinear reaction terms. The method is fourth‐order accurate in both the temporal and spatial dimensions. It requires only a regular five‐point difference stencil similar to that used in the standard second‐order algorithm, such as the Crank‐Nicolson algorithm. The Padé approximation and Richardson extrapolation are used to achieve high‐order accuracy in the spatial and temporal dimensions, respectively. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 340–354, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10012     

Long-time convergence of numerical approximations for 2D GBBM equation

We study the long-time behavior of the finite difference solution to the generalized BBM equation in two space dimensions with dirichlet boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems. Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.