中国科学院数学与系统科学研究院期刊网

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  • Jin Yan FAN
    Journal of Computational Mathematics. 2003, 21(5): 625-636.
    Based on the work of paprer [1], we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations $F(x)=0$, where $F(x):R^n\rightarrow R^n$ is continuously differentiable and $F'(x)$ is Lipschitz continuous. The algorithm is equivalent to a trust region algorithm in some sense , and the global convergence result is given. The sequence generated by the algorithm converges to the solution quadratically, if $\|F(x)\|_2$provides a local error bound for the system of nonlinear equations. Numerical results show that the algorithm performs well.
  • Tao Tang, Jiang Yang
    Journal of Computational Mathematics. 2016, 34(5): 451-461. https://doi.org/10.4208/jcm.1603-m2014-0017
    It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.
  • Fan Jiang, Deren Han, Xiaofei Zhang
    Journal of Computational Mathematics. 2018, 36(3): 351-373. https://doi.org/10.4208/jcm.1605-m2016-0828
    Tensor canonical decomposition (shorted as CANDECOMP/PARAFAC or CP) decomposes a tensor as a sum of rank-one tensors, which finds numerous applications in signal processing, hypergraph analysis, data analysis, etc. Alternating least-squares (ALS) is one of the most popular numerical algorithms for solving it. While there have been lots of efforts for enhancing its efficiency, in general its convergence can not been guaranteed.
    In this paper, we cooperate the ALS and the trust-region technique from optimization field to generate a trust-region-based alternating least-squares (TRALS) method for CP. Under mild assumptions, we prove that the whole iterative sequence generated by TRALS converges to a stationary point of CP. This thus provides a reasonable way to alleviate the swamps, the notorious phenomena of ALS that slow down the speed of the algorithm. Moreover, the trust region itself, in contrast to the regularization alternating least-squares (RALS) method, provides a self-adaptive way in choosing the parameter, which is essential for the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALS in[26], which only proved the cluster point of the iterative sequence generated by RALS is a stationary point. In order to accelerate the new algorithm, we adopt an extrapolation scheme. We apply our algorithm to the amino acid fluorescence data decomposition from chemometrics, BCM decomposition and rank-(Lr, Lr, 1) decomposition arising from signal processing, and compare it with ALS and RALS. The numerical results show that TRALS is superior to ALS and RALS, both from the number of iterations and CPU time perspectives.
  • Fu Yiping
    Journal of Computational Mathematics. 2008, 26(1): 98-111.
    In this paper, two fourth-order accurate compact difference schemes
    are presented for solving the Helmholtz equation in two space
    dimensions when the corresponding wave numbers are large. The main
    idea is to derive and to study a fourth-order accurate compact
    difference scheme whose leading truncation term, namely, the
    $\mathcal O(h^4)$ term, is independent of the wave number and the
    solution of the Helmholtz equation. The convergence property of the
    compact schemes are analyzed and the implementation of solving the
    resulting linear algebraic system based on a FFT approach is
    considered. Numerical results are presented, which support our
    theoretical predictions.
  • Dong-yang Shi, Shi-peng Mao, Shao-chun Chen
    Journal of Computational Mathematics. 2005, 23(3): 261.
    CSCD(110)
    The main aim of this paper is to study the error estimates of a nonconforming finite element with some superconvergence results under anisotropic meshes. The anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches and techniques, respectively. Furthermore, the superclose and a superconvergence estimate on the central points of elements are also obtained without the regularity assumption and quasi-uniform assumption requirement on the meshes. Finally, a numerical test is carried out, which coincides with our theoretical analysis.
  • Zhong-zhi Bai,Jun-feng Yin,Yang-feng Su
    Journal of Computational Mathematics. 2006, 24(4): 539-552.
    A shift splitting concept is introduced and, correspondingly,
    a shift-splitting iteration scheme and a shift-splitting
    preconditioner are presented,
    for solving the large sparse system of linear equations of which the
    coefficient matrix is an ill-conditioned non-Hermitian
    positive definite matrix.
    The convergence property of the shift-splitting iteration method
    and the eigenvalue distribution of the shift-splitting
    preconditioned matrix are discussed in depth,
    and the best possible choice of the shift is investigated
    in detail. Numerical computations show that
    the shift-splitting preconditioner can induce accurate, robust
    and effective preconditioned Krylov subspace iteration methods
    for solving the large sparse non-Hermitian positive definite
    systems of linear equations.
  • Jianchao Huang, Zaiwen Wen, Xiantao Xiao
    Journal of Computational Mathematics. 2017, 35(4): 529-546. https://doi.org/10.4208/jcm.1702-m2016-0699
    In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min ρ (r(x)), where r:Rn → Rm and ρ:Rm → R. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.
  • Hongkai Zhao
    Journal of Computational Mathematics. 2007, 25(4): 421-429.
    Baidu(125)
    The fast sweeping method is an efficient iterative method for
    hyperbolic
    problems.
    It combines Gauss-Seidel iterations with alternating sweeping orderings.
    In this paper several parallel implementations of the fast sweeping method
    are presented. These parallel algorithms are simple and efficient due
    to the causality of the underlying partial
    different equations. Numerical examples are used to verify our algorithms.
  • Linbo Zhang, Tao Cui, Hui Liu
    Journal of Computational Mathematics. 2009, 27(1): 89-96.
    Baidu(89)
    We present a program for computing symmetric quadrature rules on
    triangles and tetrahedra. A set of rules are obtained by using this
    program. Quadrature rules up to order 21 on triangles and up to
    order 14 on tetrahedra have been obtained which are useful for use
    in finite element computations. All rules presented here have
    positive weights with points lying within the integration domain.
  • Chunxiong Zheng
    Journal of Computational Mathematics. 2007, 25(6): 730-745.
    Baidu(32)
    In this paper we consider the numerical solution of the
    one-dimensional heat equation on unbounded domains. First an exact
    semi-discrete artificial boundary condition is derived by
    discretizing the time variable with the Crank-Nicolson method. The
    semi-discretized heat equation equipped with this boundary condition
    is then proved to be unconditionally stable, and its solution is
    shown to have second-order accuracy. In order to reduce the
    computational cost, we develop a new fast evaluation method for the
    convolution operation involved in the exact semi-discrete artificial
    boundary condition. A great advantage of this method is that the
    unconditional stability held by the semi-discretized heat equation
    is preserved. An error estimate is also given to show the dependence
    of numerical errors on the time step and the approximation accuracy
    of the convolution kernel. Finally, a simple numerical example is
    presented to validate the theoretical results.
  • Yangyang Xu
    Journal of Computational Mathematics. 2017, 35(4): 397-422. https://doi.org/10.4208/jcm.1608-m2016-0641
    Higher-order singular value decomposition (HOSVD) is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition.
    We also present one algorithm for solving the problem based on block coordinate update (BCU). Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration (HOOI) method, and thus we, for the first time, give global convergence of HOOI.
    In addition, we compare the proposed method to state-of-the-art ones for solving incomplete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.
  • Bin Wang, Xinyuan Wu, Fanwei Meng, Yonglei Fang
    Journal of Computational Mathematics. 2017, 35(6): 711-736. https://doi.org/10.4208/jcm.1611-m2016-0596
    In this paper, a novel class of exponential Fourier collocation methods (EFCMs) is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation methods. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau ⅡA methods. It turns out that the novel EFCMs are an essential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of firstorder ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM(2,2) as an illustrative example. The numerical experiments using EFCM(2,2) are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM(2,2).
  • Lie Heng WANG
    Journal of Computational Mathematics. 2003, 21(3): 321-324.
    This paper is devoted to give a new proof of Korn's inequality in LT - norm (1 < γ < ∞).
  • Ping Qi PAN
    Journal of Computational Mathematics. 2000, 18(6): 587-596.
    In this paper,we propose two new perturbation simplex variants.Solving linear programming problems without introducing artificial variables,each of the two uses the dual pivot rule to achieve primal feasibility,and then the primal pivot rule two achieve optimality.The second algorithm,a modification of the first,is designed to handle highly degenerate problems more efficiently.Some interesting results concerning merit of the perturbation are established.Numerical results from preliminary tests are also reported.
  • Jian Wang
    Journal of Computational Mathematics. 2007, 25(1): 31-048.
    In this paper, the multisymplectic Fourier pseudospectral scheme
    for initial-boundary value problems of nonlinear Schr\"{o}dinger
    equations with wave operator is considered. We investigate the
    local and global conservation properties of the multisymplectic
    discretization based on Fourier pseudospectral approximations. The
    local and global spatial conservation of energy is proved. The
    error estimates of local energy conservation law are also derived.
    Numerical experiments are presented to verify the theoretical
    predications.
  • Anping Liao, Yuan Lei
    Journal of Computational Mathematics. 2007, 25(5): 543-552.
    Baidu(19)
    Let $S_E$ denote the least-squares symmetric solution set of the
    matrix equation $AXB=C$, where A, B and C are given matrices of
    suitable size. To find the optimal approximate solution in the set
    $S_E$ to a given matrix, we give a new feasible method based on the
    projection theorem, the generalized SVD and the canonical correction
    decomposition.
  • Xiao Li, Zhonghua Qiao, Hui Zhang
    Journal of Computational Mathematics. 2017, 35(6): 693-710. https://doi.org/10.4208/jcm.1611-m2016-0517
    In this paper, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation with a variable interfacial parameter, is solved numerically by using a convex splitting scheme which is second-order in time for the non-stochastic part in combination with the CrankNicolson and the Adams-Bashforth methods. For the non-stochastic case, the unconditional energy stability is obtained in the sense that a modified energy is non-increasing. The scheme in the stochastic version is then obtained by adding the discretized stochastic term. Numerical experiments are carried out to verify the second-order convergence rate for the non-stochastic case, and to show the long-time stochastic evolutions using larger time steps.
  • Deyue Zhang , Fuming Ma
    Journal of Computational Mathematics. 2007, 25(4): 458-472.
    Consider the diffraction of a time-harmonic wave incident upon a
    periodic chiral structure. The diffraction problem may be simplified
    to a two-dimensional one. In this paper, the diffraction problem is
    solved by a finite element method with perfectly matched absorbing
    layers (PMLs). We use the PML technique to truncate the unbounded
    domain to a bounded one which attenuates the outgoing waves in the
    PML region. Our computational experiments indicate that the proposed
    method is efficient, which is capable of dealing with complicated
    chiral grating structures.
  • Helen M. Regan
    Journal of Computational Mathematics. 2002, 20(6): 611-618.
    Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. A von Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied.In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather than spatisl step size.
  • Xi Yang, Zhongqing Wang
    Journal of Computational Mathematics. 2015, 33(1): 59-85. https://doi.org/10.4208/jcm.1405-m4368
    Baidu(3)
    In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.
  • Zhongzhi Bai, Yonghua Gao
    Journal of Computational Mathematics. 2007, 25(5): 498-511.
    We construct a modified Bernoulli iteration method for solving the
    quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and
    $C$ are square matrices. This method is motivated from the
    Gauss-Seidel iteration for solving linear systems and the
    Sherman-Morrison-Woodbury formula for updating matrices. Under
    suitable conditions, we prove the local linear convergence of the
    new method. An algorithm is presented to find the solution of the
    quadratic matrix equation and some numerical results are given to
    show the feasibility and the effectiveness of the algorithm. In
    addition, we also describe and analyze the block version of the
    modified Bernoulli iteration method.
  • Qian Zhang, Ran Zhang
    Journal of Computational Mathematics. 2016, 34(5): 532-548. https://doi.org/10.4208/jcm.1604-m2015-0413
    In this paper, we present a weak Galerkin (WG) mixed finite element method for solving the second-order elliptic equations with Robin boundary conditions. Stability and a priori error estimates for the coupled method are derived. We present the optimal order error estimate for the WG-MFEM approximations in a norm that is related to the L2 for the flux and H1 for the scalar function. Also an optimal order error estimate in L2 is derived for the scalar approximation by using a duality argument. A series of numerical experiments is presented that verify our theoretical results.
  • Caixia Kou, Zhongwen Chen, Yuhong Dai, Haifei Han
    Journal of Computational Mathematics. 2018, 36(3): 331-350. https://doi.org/10.4208/jcm.1705-m2016-0820
    An augmented Lagrangian trust region method with a bi-object strategy is proposed for solving nonlinear equality constrained optimization, which falls in between penalty-type methods and penalty-free ones. At each iteration, a trial step is computed by minimizing a quadratic approximation model to the augmented Lagrangian function within a trust region. The model is a standard trust region subproblem for unconstrained optimization and hence can efficiently be solved by many existing methods. To choose the penalty parameter, an auxiliary trust region subproblem is introduced related to the constraint violation. It turns out that the penalty parameter need not be monotonically increasing and will not tend to infinity. A bi-object strategy, which is related to the objective function and the measure of constraint violation, is utilized to decide whether the trial step will be accepted or not. Global convergence of the method is established under mild assumptions. Numerical experiments are made, which illustrate the efficiency of the algorithm on various difficult situations.
  • Yuanpeng Zhu, Xuli Han
    Journal of Computational Mathematics. 2015, 33(6): 642-684. https://doi.org/10.4208/jcm.1509-m4414
    Four new trigonometric Bernstein-like basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézier-like curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C2FC3 continuous for a non-uniform knot vector, and C3 or C5 continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for G1 continuous joining two trigonometric Bézier-like patches over triangular domain are deduced.
  • Lenys Bello, Marcos Raydan
    Journal of Computational Mathematics. 2005, 23(3): 225.
    The spectral gradient method has proved to be effective for solving large-scale unconstrained optimization problems. It has been recently extended and combined with the projected gradient method for solving optimization problems on convex sets. This combination includes the use of nonmonotone line search techniques to preserve the fast local convergence. In this work we further extend the spectral choice of steplength to accept preconditioned directions when a good preconditioner is available. We present an algorithm that combines the spectral projected gradient method with preconditioning strategies to increase the local speed of convergence while keeping the global properties. We discuss implementation details for solving large-scale problems.
  • Zhong-zhi Bai,De-ren Wang,D.J. Evans
    Journal of Computational Mathematics. 1998, 16(3): 221-238.
    A class of asynchronous matrix multi-splitting multi-parameter
    relaxation methods, including the asynchronous matrix multisplitting
    SAOR, SSOR and SGS methods as well as the known asynchronous matrix
    multisplitting AOR, SOR and GS methods, etc., is proposed for solving
    the large sparse systems of linear equations by making use of the
    principle of sufficiently using the delayed information. These new
    methods can greatly execute the parallel computational efficiency of the
    MIMD-systems, and are shown to be convergent when the coefficient
    matrices are $H$-matrices. Moreover, necessary and sufficient conditions
    ensuring the convergence of these methods are concluded for the case
    that the coefficient matrices are $L$-matrices.
  • Cheng Jian ZHANG
    Journal of Computational Mathematics. 2002, 20(6): 583-590.
    This paper first presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDEs). Then the numerical analogous results, of the natural Runge-Kutta (NRK) methods for the same class of nonliner NDDEs,are given.In particular,it is shown thar the (k,l)-algebraic stability of a RK method for ODEs implies the generalized asymptotic stability and the global stability of the induced NRK method.of
  • Malte Braack, Piotr Boguslaw Mucha
    Journal of Computational Mathematics. 2014, 32(5): 507-521. https://doi.org/10.4208/jcm.1405-m4347
    The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, arti cial boundaries are necessary. A well established arti cial boundary condition for the Navier-Stokes equations discretized by finite elements is the "do-nothing" condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established out ow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a "directional do-nothing" condition. In contrast to the usual "do-nothing" condition this boundary condition has enhanced stability properties. In the case of pure out ow, the condition is equivalent to the original one, whereas in the case of in ow a dissipative e ect appears. We show existence of weak solutions and illustrate the e ect of this boundary condition by computation of steady and non-steady flows.
  • Huang Lan Chien
    Journal of Computational Mathematics. 1983, 1(1): 75-082.

    An upwind difference scheme was given by the author in [5] for the numerical solution of steady-state problems. The present work studies this upwind scheme and its corresponding boundary scheme for the numerical solution of unsteady problems. For interior points the difference equations are approximations of the characteristic relations corresponding to the outgoing characteristics and the "non-reflecting" boundary conditions. Calculation of a Riemann problem in a finite computational region yields promising numerical results.

  • Qi DUAN, Huan Ling ZHANG, Xiang LAI, Nan XIE,Fu Hua CHENG
    Journal of Computational Mathematics. 2001, 19(2): 143-150.
    Baidu(27)
    In this paper,a kind of rational cubic interpolation function with linear denominator is constructed.The constrained interpolation with constraint on shape of the interpolating curves and on the second-order derivative of the interpolating function is studied by using this interpolation,and as the consequent result,the convex interpolation conditions have been derived.
  • Zhongying Chen , Yao Lu , Yuesheng Xu , Hongqi Yang
    Journal of Computational Mathematics. 2008, 26(1): 37-55.
    Baidu(58)
    We consider solving linear ill-posed operator equations. Based on a
    multi-scale decomposition for the solution space, we propose a
    multi-parameter regularization for solving the equations. We
    establish weak and strong convergence theorems for the
    multi-parameter regularization solution. In particular, based on the
    eigenfunction decomposition, we develop a posteriori choice strategy
    for multi-parameters which gives a regularization solution with the
    optimal error bound. Several practical choices of multi-parameters
    are proposed. We also present numerical experiments to demonstrate
    the outperformance of the multi-parameter regularization over the
    single parameter regularization.
  • Weijun Zhou,Donghui Li
    Journal of Computational Mathematics. 2007, 25(1): 89-096.
    Baidu(35)
    In this paper, we propose an algorithm for solving nonlinear
    monotone equations by combining the limited memory BFGS method
    (L-BFGS) with a projection method. We show that the method is
    globally convergent if the equation involves a
    Lipschitz continuous monotone function. We also present some
    preliminary numerical
    results.
  • Bing-sheng He,Li-zhi Liao,Mai-jian Qian
    Journal of Computational Mathematics. 2006, 24(6): 693-710.
    The monotone variational inequalities VI$(\Omega,F)$ have vast applications,
    including optimal controls and convex programming. In this paper we focus
    on the VI problems that have a particular splitting structure and in which
    the mapping $F$ does not have an explicit form, therefore only its function
    values can be employed in the numerical methods for solving such problems.
    We study a set of numerical methods that are easily implementable. Each
    iteration of the proposed methods consists of two procedures. The first
    (prediction) procedure utilizes alternating projections to produce a predictor.
    The second (correction) procedure generates the new iterate via some minor
    computations. Convergence of the proposed methods is proved under mild
    conditions.
    Preliminary numerical experiments for some traffic equilibrium problems
    illustrate the effectiveness of the proposed methods.
  • Wei Wang, Lian-sheng Zhang, Yi-fan Xu
    Journal of Computational Mathematics. 2005, 23(2): 217-224.
    CSCD(2)
    A revised conjugate gradient projection method for nonlinear inequality constrained optimization problems is proposed in the paper, since the search direction is the combination of the conjugate projection gradient and the quasi-Newton direction. It has two merits. The one is that the amount of computation is lower because the gradient matrix only needs to be computed one time at each iteration. The other is that the algorithm is of global convergence and locally superlinear convergence without strict complementary condition under some mild assumptions. In addition the search direction is explicit.
  • J.-D. Benamou, Songting Luo, Hongkai Zhao
    Journal of Computational Mathematics. 2010, 28(4): 489-516. https://doi.org/10.4208/jcm.1003-m0014
    We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on:
    1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities;
    2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil.
    We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities. Numerical examples are presented.
  • Yanzhao Cao, Ran Zhang , Kai Zhang
    Journal of Computational Mathematics. 2008, 26(5): 702-715.
    In this paper, we consider the finite element method and
    discontinuous Galerkin method for the stochastic Helmholtz equation
    in $\mathbb{R}^d$ $(d=2,3)$. Convergence analysis and error
    estimates are presented for the numerical solutions. The effects of
    the noises on the accuracy of the approximations are illustrated.
    Numerical experiments are carried out to verify our theoretical
    results.
  • Sun Geng
    Journal of Computational Mathematics. 1993, 11(3): 250-260.
    CSCD(19)
    Characterizations of symmetric and symplectic Runge-Kutta methods, which are based on the W-transformation of Harier and Wanner, are presented. Using these characterizations we construct two classes of high order symplectic (symmetric and algebraically stable or algebraically stable) Runge-Kutta methods. They include and extend known classes of high order implicit Runge-Kutta methods.
  • Xiao-xia Guo, Zhong-zhi Bai
    Journal of Computational Mathematics. 2005, 23(3): 305.
    CSCD(2)
    We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired $M$-matrix solution of the quadratic matrix equation $X^2-EX-F=0$ by connecting it with the nonsymmetric algebraic Riccati equation, where $E$ is a diagonal matrix and $F$ is an $M$-matrix.
  • Ming Wang , Zhong-Ci Shi , Jinchao Xu
    Journal of Computational Mathematics. 2007, 25(4): 408-420.
    In this paper, three $n$-rectangle
    nonconforming elements are proposed with $n\ge3$. They are the extensions of
    well-known Morley element, Adini element and Bogner-Fox-Schmit element in two
    spatial dimensions to any higher dimensions respectively. These elements are
    all proved to be convergent for a model biharmonic equation in $n$ dimensions.
  • Cheng Long XU,Ben Yu GUO
    Journal of Computational Mathematics. 2002, 20(4): 413-428.
    The Laguerre Gauss-Radau interpolation is investigated. Some approximation results are obtained. As an example, the Laguerre pseudospectral scheme is constructed for the BBM equation. The stability and the convergence of proposed scheme are proved. The numerical results show the high accuracy of this approch.