Using the method of high order approximate LU dicomposition, the preconditioning matrices of high order ICCG are given and the ‖R_p‖_∞ is also estimated when P=3 and 4. The upper bounds of Cond (L_PU_P)^-1A for P=3 and 4 are obtained respectively. For different mesh sizes, the computed results of Cond (L_PU_P)^-1A are also compared with the theoretical upper bound.

On the basis of pseudo elimination (PE) approach,the algorithm of the block two stage multisplitting (TSM) interative method for linear systems of the form Ax=f is proposed,when A is block tridiagonal matrix.The resulting multisplitting pseudo elimination(MPPE) method has been tested on a Challeng-L and power PC Cluster computer.Numerical examples are also given.

A modified algebraic multigrid (AMG) method for queueing networks is presented.The method keeps the singularing of queueing networks by modifying the restriction operators.Numerical results indicate that this method is more efficient and robust than conventional AMG method.

Using a model problem and the method of high order approximate LU dicomposition, the preconditioning matrices of ICCG are given and the ‖R_p‖_∞ is also estimated. Based on these results, the upper bound of condition numbers of ICCG (s,t) when (s,t)=(1,1),(1,2) and (1,3) are given respectively. For different mesh sizes, the computed results are compared with the theoretical results of this paper.

A computational formula of the theoretical curve for the curve fitting of well testing material of infinite naturally fractured reservoir is proposed. The double error control method and the logarithm searching method are used to seek the theoretical curve. A lot of experiments show that this method is with high precision and rapid calculating speed.

The objective of this study is to develop a mathematical model for calculating the initiation and spresd of fire resulting from an attack on an urban area, and to apply the model for predicting the fire-depentdent fire damage to an assumed city. The numerical test is obtained valuable information by using the parameters of fire spread from Japanese Hiroshima.The model to be developed is included——to determine description for a simplified urban,to take account of fire spread by radiation and firebrands in terms of statistical data, and to develop a computer program to perform the numerical simulation.

This paper discusses the relationship between the iterative matrix AM^-1 of the preconditioned conjugate gradient (PCG) and the tridiagonal matrix B_s obtained when the PCG iteration terminates. The distribution of eigenvalues of B_s is analysed and, consequently, several properties of eigenvalues of the PCG iterative matrix AM^-1 are revealed.

In this paper, we are concerned only with computer simulation of the fires of buildings ignited on urban areas produced by a nuclear burst. The numerical test is obtained sorne qualitative results for the simplest model by using the parameters of fire-spread from Japanese Hiroshima.

Using a difference method of irregular meshes, a numerical model for 2-dimensional oil-water reservoir simulations is presented and its numerical results of two examples have been given when IMPES method is adopted. The numerical results of example 1 are in good agreement with the analytic solution. The number of difference meshes used for example 1 is less than that used in Petrosa's model of hybrid grid for the same example and the method given in this paper is more efficient and convenient. It is difficult to solve the problem of example 2 for the difference mehtod of rectangular meshes. However, satisfying results are also obtained when the method given here is used for the numerical solution of the example 2. The results show that this method of irregular difference meshes can be applied flexibly to probems of reservoir simulations with a complex boundary and a complex structure of stratum.

Using the concepts of element order and order matrix, some practical problems are discussed in which the traditional preconditioning methods ICCG and MICCG are adopted. If the fill-in number is fixed, why the method of ICCG(s,t) becomes the most efficient when (s,t) is successively (1,1), (1,2), (1,3), (2, 4), (3, 5),..? Why the number of iterations didn't decrease when m is larger than 3 for MICCG(m)? Is it possible to improve the fill-in method of MICCG? Is it always true that MICCG is better than ICCG? It tries to give a preliminary discussion on these problems in here. From the way of high order approximate LU decomposition, a method is introduced which improves and systematizes the ICCG and MICCG. An estimation of the condition number of ICCG is given based on the discussion of the order matrix for the error matrix. It is also pointed out that there was a trouble in selecting the parameter for MICCG. A reasonable way to select the parameter is given. Thus the number of iterations of MICCG decreases when the order of MICCG increases.

A fast method is presented to compute high order approximate inverses for multidiagonal banded matrices of strong pivot. It is based on several concepts such as the order of an element in a strong pivot matrix, the order matrix and the influence areas of elimination steps.Under the condition of strong pivot, this method can be applied to unsymmetric and indefinite matrices. Coupling with the block preconditioned conjugate gradient, it is used for the numerical solution of elliptic partial differential equations and similar problems. It is demonstrated by the computational experiments that the method reported here has a higher efficiency in computations, comparing with similar methods.

It was shown in literatures of the preconditioned conjugate gradient (PCG) that the initial guess gives little influence upon the number of PCG iterations when the stopping criterion requests to reduce the residual norm by a factor. However, the examples reported here show that using a zero initial gress or different random initial guess causes the number of PCG iterations varies seriously. Moreover, the number of iterations still varies seriously for the different parameter of the modeb when the same random initial guess is used. This variation should be avoided since it may cause confusions when different methods are compared. This paper shows that if the preconditioner is given, the series {r_k} determined uniquely by the linear systems provided a zero initial guess is used. To avoid the serious variation in the number of PCG iterations, using the zero initial guess is a good choice.

Using the method of the element order in matrices, high order approximate LU decompositions are used as the preconditioners for the computation of different experiments. High order approximate inverses are also adopted for the approximate block LU decomposition in the computation of the block-PCG for same experiments. The results show that the point-PCG of order P has a higher convergence rate than the block-PCG of order P when P=2.3,…,6. The total computational cost of the point-PCG is less than the block-PCG of Ihe same order. This illustates that the point-PCG of the P-order approximate LU decomposition is more efficient than the block-PCG of the P-order approximate inverse when the order P varies in the scope considered.

A method of high order approximate LU decomposition is adopted to give the preconditioner for the coefficient matrix of 3-D elliptic partial differential equation when the normal 7-point scheme is used. Numerical results of the preconditioned conjugate gradient (PCG) solving a tipycal problem are given.The concepts of the order, the order matrix and the P-order influence areas given in this paper can be applied to a general sparse non-singular and diagonally dominant linear system of equations. The results of different meshsize show that 3-D PCG has a superlinear convergence rate. In the case of higher orders the superlinearity of the PCG convergence rate is stronger even more. Since the number of non-zero diagonals in the preconditioner increases rapidly when the order of the PCG method increases, it is not beneficial to use the preconditioner of order P if P is larger than one. The preeonditioners of order zero and order one are recommended.