2002年, 第20卷, 第5期　刊出日期：2002-09-15

  

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  BACKWARD ERROR ANALYSIS OF SYMPLECTIC INTEGRATORS FOR LINEAR SEPARABLE HAMILTONIAN SYSTEMS

  Peter G$\ddot{o}$rtz

  Journal of Computational Mathematics. 2002, 20(5): 449-460.

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  Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge-Kutta, and Runge-Kutta-Nystr$\ddot{o}$m methods applied to the simple Hamiltonian system $\dot{p}$= -vg, $\dot{q}$= kp are studied. Some new results in connection with P-stability are presented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is th eexact solution of a perturbed Hamiltonian system at discrete at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.
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  EXPANSION OF STEP-TRANSITION OPERATOR OF MULTI-STEP METHOD AND ITS APPLICATIONS (Ⅱ)

  Yi Fa TANG

  Journal of Computational Mathematics. 2002, 20(5): 461-478.

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  We give some formulae for calculation of the expansions for (1) composition of step-transition operators (STO) of any two difference schemes (DS) for ODE's, (2) inverse operator of STO of any DS, and (3) conjugate operator of STO of any DS.
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  RATE OF CONVERGENCE OF SCHWARZ ALTERNATING METHOD FOR TIME-DEPENDENT CONVECTION-DIFFUSION PROBLEM

  Jian Wei HU(1),Cai Hua WANG(2)

  Journal of Computational Mathematics. 2002, 20(5): 479-490.

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  This paper provides a theoretical justification to a overlapping domain decomposition method applied to the solution of time-dependent convection-diffusion problems. The method is based on the partial upwind finite element scheme and the discrete strong maximum principle for steady problem. An error estimate in $L^\infty(0,T;L^\infty(\Omega))$ is obtained and the fact that convergence factor $\rho(\tau,h)\rightarrow 0$ exponentially as $\tau,h\rightarrow 0$ is also proved under some usual conditions.
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  ON THE FINITE VOLUME ELEMENT VERSION OF RITZ-VOLTERRA PROJECTION AND APPLICATIONS TO RELATED EQUATIONS

  Tie ZHANG(1),Yan Ping Li(2),Robert J.Tait(2)

  Journal of Computational Mathematics. 2002, 20(5): 491-504.

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  In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal $L_2$ and $H^1$ norm error estimates, and the $L_\infty$ and $W^1_\infty$ norm error estimates by means of the time dependent Green functions. Our disc ussions also include elliptic and parabolic problems as the special cases.
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  MULTISCALE ASYMPTOTIC EXPANSION FOR A CLASS OF HYPERBOLIC-PARABOLIC TYPE EQUATION WITH HIGHLY OSCILLATORY COEFFICIENTS

  Li Qun CAO(1),De Chao ZHU(2),Jian Lan LUO(3)

  Journal of Computational Mathematics. 2002, 20(5): 505-518.

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  In this paper, we will discuss the asymptotic behaviour for a class of hyperbolic -parabolic type equation with highly oscillatory coefficients arising from the strong-transient heat and mass transfer problems of composite media. A complete multiscale asmptotic expansion ad its rigorous verification will be reported.
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  A REVERSE ORDER IMPLICIT Q-THEOREM AND THE ARNOLDI PROCESS

  Gui Zhi CHEN(1),Zhong Xiao JIA(2)

  Journal of Computational Mathematics. 2002, 20(5): 519-524.

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  Let A be a real square matrix and VTAV = G be an upper Hessenberg matrix with positive subdiagonal entries, where V is an orthogonal matrix. Then the implicit Q-theorem states that once the first column of V is given then V and G are uniquely determined too. Second, it is proved that for a Krylov subspace tewo formulations of the Arnoldi process are equivalent and in one to one correspondence. Finally, by the equivalence relation and the reverse vector sequence generated by the Arnoldi process is given, then the vector sequence and resulting Hessenberg matrix are uniquely determined.
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  ORDER RESULTS OF GENERAL LINEAR METHODS FOR MULTIPLY STIFF SINGULAR PERTURBATION PROBLEMS

  Si Qing GAN(1),Geng SUN(2)

  Journal of Computational Mathematics. 2002, 20(5): 525-532.

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  In this paper we analyze the error behavior of general linear methods applied to some classes of one-parameter multiply stiff singularly perturbed problems. We obtain the global error estimate of algebraically and diagonally stable general linear methods.The main result of this paper can be viewed as an extension of that obtained by Xiao [13] for the case of Runge-Kutta methods.
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  A NEW SMOOTHING APPROXIMATION METHOD FOR SOLVING BOX CONSTRAINED VARIATIONAL INEQUALITIES

  Chang Feng MA(1),Guo Ping LIANG(2),Shao Peng LIU(2)

  Journal of Computational Mathematics. 2002, 20(5): 533-542.

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  In this paper, we first give a smoothing approximation function of nonsmooth system based on box constrained variational inequalities and then present a new smoothing approximation algorithm. Under suitable conditions,we show that the method is globally and superoinearly convergent. Afew numerical results are also reported in the paper.
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  LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF EXPLICIT DIFFERENCE SCHEME FOR SEMILINEAR PARABOLIC EQUATIONS

  Hui FENG(1),Long Jun SHEN(2)

  Journal of Computational Mathematics. 2002, 20(5): 543-550.

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  In this paper we prove that the solution of explicit difference scheme for a class of semilinear parabolic equations converges to the solution of difference schemes for the corresponding nonlinear elliptic equations in $H^1$ norm as $t\rightarrow\infty$.We get the long time asymptotic behavior of the discrete solutions which is interested in comparting to the case of continuous solutions.
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  SPECTRAL ANALYSIS OF THE FIRST-ORDER HERMITE CUBIC SPLINE COLLOCATION DIFFERENTIATION MATRICES

  Ji Ming WU,Long Jun SHEN

  Journal of Computational Mathematics. 2002, 20(5): 551-560.

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  It has been observed numerically in [1] that, under certain conditions, all eigenvalues of the first-order Hermite cubic spline collocation differentiation matrices with unsymmetrical collocation points lie in one of the half complex planes. In this paper, we provide a theoretical proof for this spectral result.