一类无结构三角网上抛物方程的有限差分区域分解算法

本文讨论了一类在无结构三角网上数值求解二维热传导方程的有限差分区域分解算法．在这个算法中,将通过引进两类不同类型的内界点,将求解区域分裂成若干子区域．一旦内界点处的值被计算出来,其余子区域上的计算可完全并行．本文得到了稳定性条件和最大模误差估计,它表明我们的格式有令人满意的稳定性和较高的收敛阶．     

Lagrangian乘子区域分解法的一类预条件子

1．引言非重叠区域分解的Lagrangian乘子法已被许多作者讨论［1今它是一类非协调区域分解法（与通常的非协调元区域分解不同），特别适合于非匹配网格的情形（即相邻子域在公共边或公共面上的结点不重合，参见14］［6］）．这种方法的一个最大优点是不要求界面变量在内交点（或内交边）上的连续性，从而界面方程易于建立，程序易于实现，而又正因为这个特点，使得界面矩阵的预条件子不能按通常的方法构造，故目前还未见到理想的预条件子（或者条件数差，或者应用上不方便）．本文在很大程度上解决了这一问题．1）工作单位：湘潭大学数学系…     

求解线性互补问题的乘性Schwarz算法的收敛速度估计

1．引言区域分解法是八十年代兴起并得到迅速发展及广泛应用的数值计算方法．和多重网格法一样，区域分解法用于求解椭圆边值问题时具有与剖分网格h无关的收敛速度［8］，因而是一种高效快速算法．八十年代末及九十年代初，这种区域分解思想也开始应用于障碍问题的求解［2－8，10。12，16］数值实验表明，该算法对于障碍问题也是有效的·但是，和多重网格法一样，用于求解障碍问题时，算法的收敛速度分析存在一定的困难［11，13，14］对于障碍问题，一般的收敛性证明都是建立在证明算法产生的序列为一个极小化序列的基础之上［‘，‘’，“…     

基于几何非协调分解的Lagrange乘子区域分解方法

本文考虑将Lagrange乘子区域分解方法应用于几何非协调分解的情况来求解二阶椭圆问题.由于采用几何非协调区域分解,每个局部乘子空间关联到多个界面,我们按照一定的规则选取合适的乘子面来定义乘子空间.利用局部正则化技巧,可以消去内部变量,得到关于Lagrange乘子的界面方程.采用一种经济的预条件迭代方法求解界面方程,且相关的预条件子是可扩展的.     

扩散问题的一类带有加权系数的隐格式及多子域并行算法

针对扩散问题提出了一类带有加权系数的隐格式,采用分组显式和区域分解思想,又构造了若干分组显式格式.结合初边值条件,建立了求解扩散问题的一种多子域并行算法.虽然格式是隐式的,但在算法实现过程中可显式且并行地计算,这样避免了求解线性方程组的复杂性.并且当加权系数1≤θ≤2.4时,格式是无条件稳定的;0θ1时,趋向于1的方向,格式也是无条件稳定的;θ=2时,算法收敛的最快,收敛速率接近于2.通过数值试验证明此类隐格式和并行算法是有效的,计算速度快,精确度高,易于实现并行.     

含非线性源项的变分不等式的加性区域分解法

本文研究一类求解非线性变分不等式的加性区域分解法,其中区域分解为非重叠子区域,在界面上采用Robin条件,得到了算法的收敛性,而且数值算例表明,选取适合的Robin参数可加快算法的收敛速度.     

区域分解界面预条件子构造的一般框架

1．引言考虑模型问题：其中ΩR2是多边形区域,常数n≥0．将Ω作非重叠区域分解：Ω＝假定：（i）当i≠j时,（ii）当Ωi与Ωj相邻时,是Ωi和Ωj的一条公共边记称为界面）;（iii）每个闪的尺寸为d,即存在常数co和q,使出包含（包含在）一个直径为C（）（Cod）的圆（国内）．非重叠区域分解方法的实质是,引进两个变量：内部变量。h和界面变量～．先在几上并行未解子问题,将。。消去（即用～表示）,得到～的方程（称为界面方程）;再求解界面方程,得到～的值;最后将～回代,得到。人的值（即原问题的解）．这类区域分解方法是否比重…     

Helmholtz问题的Robin型区域分解法

Helmholtz问题的数值模拟在科学工程计算领域有着广泛的应用,快速高效求解Helmholtz方程离散代数系统一直是科学计算的重要研究方向.本文简要回顾了Helmholtz方程的区域分解型求解器的发展历程,重点介绍了我们提出的Robin型区域分解算法,同时比较了各类算法的优劣和特点.近年来Helmholtz方程的求解效率有了极大的提升,然而仍有一些本质困难尚待突破,如何高效求解Helmholtz方程,仍是具有挑战意义的研究课题.     

解单障碍问题的非重叠区域分解法

1．引言在实际中的许多物理问题、工程问题以及各类经济平衡问题都可用变分不等式来描述．本文考虑这类问题的数值解．众所周知，区域分解法的思想可朔源到19世纪70年代提出的Schwarz交替法，但直到本世纪中期才用于数值计算．而真正获得发展还是在近十几年．由于并行机与并行算法的发展，使得Schwarz算法的优良并行性能得以开发利用，从而使得这种区域分解新技术不仅应用于偏微分方程数值解，而且广泛应用于其它各类科学与工程计算问题．近几年来，重叠型区域分解已经被成功地应用于求解椭圆型变分不等式，早期的结果见［6］．我们还可从…     

结构三角网上抛物方程的有限差分区域分解算法

1引言对于大型科学与工程计算问题,并行计算是必需的．构造高效率的数值并行方法一直是人们关心的问题,并且已有了大量的研究．在三层交替计算方法的研究中出现了许多既具有明显并行性又绝对稳定的差分格式(见[1]-[5])．在只涉及两个时间层的算法研究中,Dawson等人(见[6])首先发展了求解一维热传导方程的区域分解算法,并将其推广到     

Parallel algorithm for the solutions of PDEs in linux clustered workstations

In this paper we propose parallel algorithm for the solution of partial differential equations over a rectangular domain using the Crank–Nicholson method by cooperation with the DuFort–Frankel method and apply it on a model problem, namely, the heat conduction equation. One of the well known parallel techniques in solving partial differential equations in cluster computing environment is the domain decomposition technique. Using this technique, the whole domain is decomposed into subdomains, each of them has its own boundaries that are called the interface points. Parallelization is realized by approximating interface values using the unconditionally stable DuFort–Frankel explicit scheme, and these values serve as Neumann boundary conditions for the Crank–Nicholson implicit scheme in the subdomains. The numerical results show that our algorithm is more accurate than the algorithm based on the forward explicit method to approximate the values of the interface points, especially, when we use a small number of time steps. Moreover, these numerical results show that increasing the number of processors which are used in the cluster, yields an increase in the algorithm speedup.     

Jacobian elliptic function method for nonlinear differential-difference equations

An algorithm is devised to derive exact travelling wave solutions of differential-difference equations by means of Jacobian elliptic function. For illustration, we apply this method to solve the discrete nonlinear Schrödinger equation, the discretized mKdV lattice equation and the Hybrid lattice equation. Some explicit and exact travelling wave solutions such as Jacobian doubly periodic solutions, kink-type solitary wave solutions are constructed.     

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first‐ and second‐order splitting up method, of additive type, for the dependent variables is initially considered to solve the two‐ or three‐dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second‐order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A mixed finite element method with Lagrange multipliers for nonlinear exterior transmission problems

Summary. We apply a mixed finite element method to numerically solve a class of nonlinear exterior transmission problems in R ² with inhomogeneous interface conditions. Besides the usual unknowns required for the dual-mixed method, which include the gradient of the temperature in this nonlinear case, our approach makes use of the trace of the outer solution on the transmission boundary as a suitable Lagrange multiplier. In addition, we use a boundary integral operator to reduce the original transmission problem on the unbounded region into a nonlocal one on a bounded domain. In this way, we are lead to a two-fold saddle point operator equation as the resulting variational formulation. We prove that the continuous formulation and the associated Galerkin scheme defined with Raviart-Thomas spaces are well posed, and derive the a-priori estimates and the corresponding rate of convergence. Then, we introduce suitable local problems and deduce first an implicit reliable and quasi-efficient a-posteriori error estimate, and then a fully explicit reliable one. Finally, several numerical results illustrate the effectivity of the explicit estimate for the adaptive computation of the discrete solutions. Mathematics Subject Classification (2000): 65N30, 65N38, 65N22, 65F10This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.     

Lie symmetry analysis to Fisher's equation with time fractional order

The aim of this letter is to apply the Lie group analysis method to the Fisher''s equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher''s(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher''s is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.     

Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces

We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.     

The spectral methods for parabolic Volterra integro-differential equations

In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.     

关于矩阵族稳定的(J)条件及其与Buchanan准则的关系

本文注意到矩阵族稳定的Kreiss定理和Buchanan准则不便于实际应用。文中(§2,§3)从Kreiss定理的豫解条件出发得到了至少对于四阶以下矩阵族较为实用的判别稳定性的(J)条件;并证明了对于其特征值赋套的上三角矩阵族,(J)条件与Buchanan准则的等价性。§4作为(J)条件的应用讨论了逼近于二维、三维波动方程的显式差分方程(其增长矩阵分别是三、四阶矩阵族),得到了稳定的充要条件。     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.