一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

轴对称Poisson方程的Trefftz有限元解法

将径向基函数应用到一类轴对称Poisson方程的数值求解中,提出了一种Trefftz有限元计算格式.非0右端项将问题的特解引入Trefftz单元域内场,致使单元刚度方程涉及区域积分.利用径向基函数对特解近似处理,可消除区域积分,从而保持Trefftz有限元法只含边界积分的优势.为获得特解,选取求解域内所有单元的节点和形心作为基本插值点,而在求解域之外构造一个虚拟边界,在其上布置一定数目的虚拟点作为额外插值点.数值算例验证了该方法的有效性和可行性.     

三维重调和方程的双方程边界积分方程法

以守恒积分为工具,推导了三维重调和方程的新的边界积分方程,所得出的新方程与传统的边界积分方程相比较,降低了奇异性,避免了传统边界元方法中的强奇异积分的计算.对不同边界都采用第二类积分方程,得到了三维重调和方程的双方程方法.     

无穷凹角区域各向异性问题的自然边界元法

本文研究无穷凹角区域上一类各向异性问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式和自然积分方程,给出了自然积分方程的数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性.     

改进的无奇异局部边界积分方程方法

在局部边界积分方程方法中,当源节点位于分析域的整体边界上时,局部边界积分将出现奇异积分问题,这些奇异积分需要做特别的处理．为此,提出了对域内节点采用局部积分方程,而对边界节点直接采用移动最小二乘近似函数引入边界条件来解决奇异积分问题,这同时也解决了对积分边界进行插值引入近似误差的问题．作为应用和数值实验,对Laplace方程和Helmholtz方程问题进行了分析,取得了很好的数值结果．进而,在Helmholtz方程求解中,采用了含波解信息的修正基函数来代替单项式基函数进行近似．数值结果显示,这样处理是简单高效的,在高波数声传播问题的求解中非常具有前景．     

弹性力学平面问题的等价的边界积分方程

严格地推导出与弹性力学平面问题的微分方程边值问题等价的间接和直接未知量边界积分方程,用实例指出某些习用的边界积分方程有时不必要或不充分,并进行了数值比较.     

椭圆外区域上Helmholtz问题的自然边界元法

本文研究椭圆外区域上Helmholtz方程边值问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式及自然积分方程,给出了自然积分方程的数值方法.由于计算的需要,我们详细地讨论了Mathieu函数的计算方法(当0相似文献   

Helmholtz方程Cauchy问题的间接积分方程方法

 本文处理二维和三维Helmholtz方程的边界数据复原问题.通过利用位势理论近似问题的解,导出了解决Cauchy问题的一种非迭代积分方程方法.为了处理形成问题的不适定性,采用了Tikhonov正则化结合Morozov偏差原理的方法,并且给出了算法的收敛性和误差估计,最后给出了二维和三维的数值算例.通过数值算例我们检验了源点和边界之间距离的关系,算法关于噪声、源点数目的数值收敛性,稳定性.     

海面下方多障碍体散射问题数值研究

本文针对二维空间中海面下方多障碍体散射问题,分别从理论分析和数值计算两方面进行研究.通过分析散射问题的特性,利用Helmholtz方程,结合不同边界条件以及无穷远处辐射条件,建立了海面下方多障碍体散射问题的数学模型,并证明了散射问题解的唯一性.基于位势理论,利用间接积分方程方法,得到了不同区域的场所满足的积分表示,以及边界上密度函数所满足的边界积分方程.通过引入位势算子,将积分区域进行截断,得到有界域上的算子方程.针对所建立的边界积分方程系统,利用Nystr?m方法构造数值格式,并证明了数值解的收敛性.最后,利用数值实验验证理论的正确性和有效性.进一步,通过设计数值实验分析不同参数对散射问题的影响.     

Modified High-order Upwind Method for Convection Diffusion Equation

Abstract In this paper, we study the high-order upwind finite difference method for steady convection-diffusionproblems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference schemeon nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is inconversation form, satisfies maximum value principle and has second-order error estimates in discrete H~1 norm.To illustrate our conclusion, several numerical examples are given.     

A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

Iterative numerical schemes for solving the electrostatic partial differential equation with variable conductivity, using fast and high-order accurate direct methods for preconditioning, are compared. Two integral method schemes of this type, based on previously suggested splittings of the equation, are proposed, analyzed, and implemented. The schemes are tested for large problems on a square. Particular emphasis is paid to convergence as a function of geometric complexity in the conductivity. Differences in performance of the schemes are predicted and observed in a striking manner. AMS subject classification (2000) 31A10, 35C15, 65R20.Received May 2004. Accepted September 2004. Communicated by Anders Szepessy.Johan Helsing: This work was supported by the Swedish Science Research Council under contract 621-2001-2799.     

Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition

We consider the numerical solution for the Helmholtz equation in R~2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.     

Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.     

Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime

As is known, the numerical stiffness arising from the small mean free path is one of the main difficulties in the kinetic equations. In this paper, we derive both the split and the unsplit schemes for the linear semiconductor Boltzmann equation with a diffusive scaling. In the two schemes, the anisotropic collision operator is realized by the "BGK"-penalty method, which is proposed by Filbet and Jin [F. Filbet and S. Jin, J. Comp. Phys. 229(20), 7625-7648, 2010] for the kinetic equations and the related problems having stiff sources. According to the numerical results, both of the schemes are shown to be uniformly convergent and asymptotic-preserving. Besides, numerical evidences suggest that the unsplit scheme has a better numerical stability than the split scheme.     

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper, a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this approach. We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation, which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.AMS subject classifications: 65M70, 41A30, 81Q05     

Modified High-order Upwing Method for Convection Diffusion Equation

Abstract   In this paper, we study the high-order upwind finite difference method for steady convection-diffusion problems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference scheme on nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is in conversation form, satisfies maximum value principle and has second-order error estimates in discrete H ¹ norm. To illustrate our conclusion, several numerical examples are given. Supported by the Research Fund for Doctoral Program of High Education of China State Education Commission.     

A Switching Algorithm Based on Modified Quasi-Newton Equation

In this paper, a switching method for unconstrained minimization is proposed. The method is based on the modified BFGS method and the modified SR1 method. The eigenvalues and condition numbers of both the modified updates are evaluated and used in the switching rule. When the condition number of the modified SR1 update is superior to the modified BFGS update, the step in the proposed quasi-Newton method is the modified SR1 step. Otherwise the step is the modified BFGS step. The efficiency of the proposed method is tested by numerical experiments on small, medium and large scale optimization. The numerical results are reported and analyzed to show the superiority of the proposed method.     

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an $h$-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.     

Direct Implementation of Tikhonov Regularization for the First Kind Integral Equation

A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.