三维对流扩散问题沿特征线多步离散Galerkin法及交替方向预处理迭代解

对三维依赖时间对流扩散问题构造了沿特征方向多步离散Galerkin格式 ,并用交替方向预处理迭代法解沿特征线多步离散Galerkin法在每一时间步所产生的代数方程组 .给出了迭代解的最优L2 模误差估计以及此方法的几乎是最优的工作量估计 .     

Modified nodal cubic spline collocation for biharmonic equations

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N ² log₂ N + mN ²) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log₂ N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.      

一类改进的高斯-赛德尔迭代法的比较性定理

讨论了改进的高斯-赛德尔迭代法的收敛性．若系数矩阵为非奇异不可约M-矩阵。则该预条件下高斯-赛德尔迭代法收敛的快慢取决于原高斯-赛德尔迭代法谱半径的大小．同样,在该预条件下高斯-赛德尔迭代法的谱半径大小与其他高斯-赛德尔迭代法的谱半径大小有关     

新的预条件AOR迭代方法和比较定理

提出了一种新的预条件AOR迭代法,对其收敛性进行了分析,给出该预条件AOR迭代法与经典AOR迭代法之间的比较性定理.最后的数值例子说明该预条件要优于经典的AOR迭代法.     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

High-efficiency improved symmetric successive over-relaxation preconditioned conjugate gradient method for solving large-scale finite element linear equations

Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.     

求解时谐涡流模型鞍点问题的分块交替分裂隐式迭代算法的改进

分块交替分裂隐式迭代方法是求解具有鞍点结构的复线性代数方程组的一类高效迭代法.本文通过预处理技巧得到原方法的一种加速改进方法,称之为预处理分块交替分裂隐式迭代方法·理论分析给出了新方法的收敛性结果.对于一类时谐涡旋电流模型问题,我们给出了若干满足收敛条件的迭代格式.数值实验验证了新型算法是对原方法的有效改进.     

关于Z-矩阵的一类新预条件迭代法

 给出了解线性方程组Ax=b的一类新的预条件迭代法,并证明了其收敛性.数值例子表明,所给方法比经典的Gauss-Seidel方法收敛速度快.     

Efficient planewise‐like preconditioners for solving 3D problems

We deal with the numerical solution of large linear systems resulting from discretizations of three‐dimensional boundary value problems. It has been shown recently that, if the use of presently available planewise pre‐conditionings is as pathological as thought by many people, except for some trivial anisotropic problems, linewise preconditionings could fairly outperform pointwise methods of approximately the same computational complexity. We propose here a zebra (or line red–black) like numbering strategy of the grid points that leads to a rate of convergence comparable to the one predicted for ideal planewise preconditionings. The keys to the success of this strategy are threefold. On the one hand, one gets rid of the, time and memory consuming, task of computing some accurate approximation to the inverse of each pivot plane matrix. On the other hand, at each PCG iteration, there is no longer a need to solve linear systems whose matrices have the same structure as a two‐dimensional boundary value problem matrix. Finally, it is well suited to parallel computations. Copyright © 1999 John Wiley & Sons, Ltd.     

Improved preconditioned conjugate gradient method and its application in F. E. A. for engineering

In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method(PCCG).The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix.The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix.This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems,and simultaneously contrasted with other methods.The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations,It is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.