Symmetric modified AOR method to solve systems of linear equations
Mohammad Taghi Darvishi, Farzad Khani, Ali Mohammad Godarzi and Hongjiong Tian
J. Appl. Math. Comp., DOI: 10.1007/s12190-010-0387-6
We propose a class of symmetric modified accelerated overrelaxation (SMAOR) methods for solving large sparse linear systems. The convergence region of the method has been investigated. Numerical examples indicate that the SMAOR method is better than other methods such as accelerated overrelaxation (AOR) and modified accelerated overrelaxation (MAOR) methods, since the spectral radius of iteration matrix in SMAOR method is less than that of the other methods. Also, we apply the method to solve a real boundary value problem.
Spectral method on quadrilaterals
Benyu Guo and Hongli Jia
Math. Comp., 79(2010), 2237-2264.
In this paper, we investigate the spectral method on quadrilaterals. We introduce an orthogonal family of functions induced by Legendre polynomials, and establish some results on the corresponding orthogonal approximation. These results play important roles in the spectral method for partial differential equations defined on quadrilaterals. As examples of applications, we provide spectral schemes for two model problems and prove their spectral accuracy in Jacobi weighted Sobolev space. Numerical results coincide well with the analysis. We also investigate the spectral method on convex polygons whose solutions possess spectral accuracy. The approximation results of this paper are also applicable to other problems.
Composite Laguerre-Legendre spectral method for exterior problems
Benyu Guo and Tianjun Wang
Adv. Comp. Math., 32(2010), 393-429.
In this paper, we propose a composite Laguerre-Legendre spectral method for two-dimensional exterior problems. Results on the composite Laguerre-Legendre approximation, which is a set of piecewise mixed approximations coupled with domain decomposition, are established. These results play important roles in the related spectral methods for exterior problems. As examples of applications, the composite spectral schemes are provided for two model problems, with the convergence analysis. An efficient implementation is described. Numerical results demonstrate the spectral accuracy in space.
Composite Laguerre-Legendre spectral method for fourth-order exterior problems
Benyu Guo and Tianjun Wang
J. Sci. Comp., 44(2010), 255-285.
In this paper, we investigate composite Laguerre-Legendre spectral method for fourth-order exterior problems. Some results on composite Laguerre-Legendre approximation are established, which is a set of piecewise mixed approximations coupled with domain decomposition. These results play an important role in spectral method for fourth-order exterior problems with rectangle obstacle. As examples of applications, composite spectral schemes are provided for two model problems, with convergence analysis. Efficient algorithms are implemented. Numerical results demonstrate their high accuracy, and confirm theoretical analysis well.
A spectral collocation method for solving initial value problems of first order ordinary differential equations
Benyu Guo and Zhongqing Wang
Disc. Cont. Dyna. Syst. B, 14(2010), 1029-1054.
We propose a spectral collocation method for solving initial value problems of first order ODEs, based on the Legendre-Gauss-Lobatto interpolation. This method is easy to be implemented and possesses the spectral accuracy. We also develop a multi-step version of this process, which is very available for long-time calculation. Numerical results demonstrate the high accuracy of suggested algorithms and coincide well with the theoretical analysis.
Generalized Jacobi rational spectral method and its applications
Benyu Guo and Yonggang Yi
J. Sci. Comp., 43(2010), 201-238.
We introduce an orthogonal system on the whole line, induced by the generalized Jacobi functions. Some results on the generalized Jacobi rational approximation are established, which play important roles in the related spectral methods. As examples of applications, the rational spectral schemes are proposed for sine-Gordon, Klein-Gordon and Fisher equations, with the convergence analysis. Numerical results demonstrate their efficiency.
Dynamical analysis of Kirchhoff plates by an explicit time integration Morley element method
Ling Guo and Jianguo Huang
J. Comp. Appl. Math., 234(2010), 2483-2492.
An explicit time integration finite element method is proposed to investigate dynamical analysis of Kirchhoff plates, where the Morley element is used for spatial discretization and the second-order central scheme for time discretization. Certain error estimates in the energy norm are achieved. A number of numerical results are included to show computational performance of the method.
Adini Q1-P3 FEM for general elastic multi-structure problems
Ling Guo and Jianguo Huang
Numer. Meth. Part. Diff. Equa., 2010, DOI 10.1002/num.20571.
An Adini-Q1-P3 finite element method is introduced to solve general elastic multi-structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, and transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples.
Discrete-ordinate discontinuous Galerkin methods fo solving the radiative transfer equation
Weimin Han, Jianguo Huang and Joseph A. Eichholtz
SIAM J. Sci. Comp., 32(2010), 477-497.
The radiative transfer equation (RTE) occurs in a wide variety of applications. In this paper, we study discrete-ordinate discontinuous Galerkin methods for solving the RTE. The numerical methods are formed in two steps. In the first step, the discrete ordinate technique is applied to discretize the integral operator for the angular variable, resulting in a semi-discrete hyperbolic system. In the second step, the spatial discontinuous Galerkin method is applied to discretize the semi-discrete system. A stability and error analysis is performed on the numerical methods. Some numerical examples are included to demonstrate the convergence behavior of the methods.
A new C discontinuous Galerkin method for Kirchhoff plates
Jianguo Huang, Xuehai Huang and Weimin Han
Comp. Meth. Appl. Mech. Engr., 199(2010), 1446-1454.
A general framework of constructing C discontinuous Galerkin (CDG) methods is developed for solving the Kirchhoff plate bending problem, following some ideas by Cockburn. The numerical traces are determined based on a discrete stability identity, which lead to a class of stable CDG methods. A stable CDG method, called the LCDG method, is particularly interesting in our study. It can be viewed as an extension to fourth-order problems of the LDG method. For this method, optimal order error estimates in certain broken energy norm and -norm are established. Some numerical results are reported, confirming the theoretical convergence orders.
An iterative algorithm for solving a finite-dimensional linear operator
equation T(x)=f with applications
JianguoHuang and Liwei Nong
Linear Alge. Appl., 432(2010), 1176-1188.
This paper proposes an iterative algorithm for solving a general finite-dimensional linear operator equation T(x)=f and demonstrates that it will get the exact solution within a finite number of iteration steps. This algorithm unifies all the iterative methods in several papers and provides an iterative method for solving an inverse problem related to Hermitian-generalized Hamiltonian matrices.
Vibration analysis for elastic multi-beam structures by the C0-continuous time-stepping finite element method
Junjiang Lai, Jianguo Huang and Zhongci Shi
Int. J. Numer. Meth. Biomed. Engr., 26(2010), 205-233.
Some C0-continuous time-stepping finite element method is proposed for investigating vibration analysis of elastic multi-beam structures. In the time direction, the C0-continuous Galerkin method is used to discretize the generalized displacement field. In the space directions, the longitudinal displacements and rotational angles on beams are discretized using conforming linear elements, while the transverse displacements on beams are discretized by the Hermite elements of third order. The error of the method in the energy norm is proved to be O(h+k3), where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. The finite difference analysis in time is developed to discuss the spectral behavior of the algorithms as well as their dissipation and dispersion properties in the low-frequency regime. The method has also been extended to study some nonlinear problems. A number of numerical tests are included to illustrate the computational performance of the method.
A lumped mass finite element method for vibration analysis of elastic plate-plate structures
Junjiang Lai, Jianguo Huang and Zhongci Shi
Sci. China Ser. A, 53(2010), 1453-1474.
The fully discrete lumped mass finite element method is proposed for vibration analysis of elastic plate-plate structures. In the space directions, the longitudinal displacements on plates are discretized by conforming linear elements, and the transverse displacements are discretized by the Morley element. By means of the second order central difference for discretizing the time derivative and the technique of lumped masses, a fully discrete lumped mass finite element method is obtained, and two approaches to choosing the initial functions are also introduced. The error analysis for the method in the energy norm is established, and some numerical examples are included to validate the theoretical analysis.
An efficient control variate method for pricing variance derivatives
Junmei Ma and Chenglong Xu
J. Comp. Appl. Math., 235 (2010), 108-119.
This paper studies the pricing of variance swap derivatives with stochastic volatility by the control variate method. A closed form solution is derived for the approximate model with deterministic volatility, which plays the key role in the paper, and an efficient control variate technique is therefore proposed when the volatility obeys the log-normal process. By the analysis of moments for the underlying processes, the optimal volatility function in the approximate model is constructed. The numerical results show the high efficiency of our method; the results coincide with the theoretical results. The idea in the paper is also applicable for the valuation of other types of variance swap, options with stochastic volatility and other financial derivatives with multi-factor models.
Nonlinear fourth-order elliptic equations with nonlocal boundary conditions
Chia Ven Pao and Yuanming Wang
J. Math. Anal. Appl., 372(2010), 351–365.
This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The Monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.
A mathematical model of prostate tumor growth under hormone therapy with mutation inhibitor
Youshan Tao, Qian Guo and Kazuyuki Aihara
J. Nonli. Sci., 20(2010), 219–240.
This paper extends Jackson’s model describing the growth of a prostate tumor with hormone therapy to a new one with hypothetical mutation inhibitors. The new model not only considers the mutation by which androgen-dependent (AD) tumor cells mutate into androgen-independent (AI) ones but also introduces inhibition which is assumed to change the mutation rate. The tumor consists of two types of cells (AD and AI) whose proliferation and apoptosis rates are functions of androgen concentration. The mathematical model represents a free-boundary problem for a nonlinear system of parabolic equations, which describe the evolution of the populations of the above two types of tumor cells. The tumor surface is a free boundary, whose velocity is equal to the cell’s velocity there. Global existence and uniqueness of solutions of this model is proved. Furthermore, explicit formulae of tumor volume at any time t are found in androgen-deprived environment under the assumption of radial symmetry, and therefore the dynamics of tumor growth under androgen-deprived therapy could be predicted by these formulae. Qualitative analysis and numerical simulation show that controlling the mutation may improve the effect of hormone therapy or delay a tumor relapse.
A numerical method for solving the inverse heat conduction problem without initial value
Yanbin Wang, Jin Cheng, Junichi Nakagawa and Masahiro Yamamoto
Inv. Prob. Sci. Eng. 18 (2010), 655–671.
We consider the inverse heat conduction problem for the one-dimensional heat equation, where we are requested to determine a boundary value at one end of a spatial interval over a time interval and an initial value by means of Cauchy data at another end. By the existing theory we can prove the uniqueness in determining both a boundary value and an initial value, and our method does not require any initial value. We test our numerical method and show stable numerical reconstruction.
On 2nth-order nonlinear multi-point boundary value problems
Math. Comp. Model., 51 (2010), 1251–1259.
This paper is concerned with the existence and uniqueness of a solution for a class of 2nth-order nonlinear multi-point boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone condition on the nonlinear function. A sufficient condition for the uniqueness of a solution is given. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. Two examples are presented to illustrate the results. Some numerical results are given. All the conclusions are directly applicable to the finite difference solution of the corresponding ordinary differential system.
The iterative solutions of 2nth-order nonlinear multi-point boundary value problems
Appl. Math. Comp., 217 (2010), 2251–2259.
The aim of this paper is to investigate the existence of iterative solutions for a class of 2nth-order nonlinear multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as three- or four-point boundary condition, (n+2)-point boundary condition and 2(n- m)-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique. A monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution. Two examples are presented to illustrate the results.
Numerical solutions of a nonlinear reaction-diffusion system
Yuanming Wang and Yuan Gong
Int. J. Comp. Math., 87 (2010), 1975–2002.
This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.
A collocation method with exact imposition of mixed boundary conditions
Zhongqing Wang and Lilian Wang
J. Sci. Comp., 42(2010), 291-317.
In this paper, we propose a natural collocation method with exact imposition of mixed boundary conditions based on a generalized Gauss-Lobatto-Legendre-Birhoff quadrature rule that builds in the underlying boundary data. We provide a direct construction of the quadrature rule, and show that the collocation method can be implemented as efficiently as the usual collocation scheme for PDEs with Dirichlet boundary conditions. We apply the collocation method to some model PDEs and the time-harmonic Helmholtz equation, and demonstrate its spectral accuracy and efficiency by various numerical examples.
A Legendre-Gauss collocation method for nonlinear delay differential equations
Zhongqing Wang and Lilian Wang
Disc. Cont. Dyna. Syst. B, 13(2010), 685-708.
In this paper, we introduce an efficient Legendre-Gauss collocation method for solving nonlinear delay differential equations with variable delay. We analyze the convergence of the single-step and multi-domain versions of the proposed method, and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. We also make numerical comparison with other methods.
Spherical harmonic-generalized Laguerre pseudospectral method for three dimensional exterior problems
Zhongqing Wang, Rong Zhang and Benyu Guo
Inter. J. Comp. Math., 87(2010), 2123-2142.
In this paper, we develop the mixed pseudospectral method for three-dimensional exterior problems. Some basic results on the mixed spherical harmonic-generalized Laguerre interpolation are established, which play important roles in the related pseudospectral methods. As examples, we provide the mixed pseudospectral schemes for two exterior problems with convergence analysis. Numerical results demonstrate the efficiency of this approach.
Generalized Hermite spectral method and its applications to problems in unbounded domains
Xinmin Xiang and Zhongqing Wang
SIAM J. Numer. Anal., 48(2010), 1231-1253.
In this paper, we develop a spectral method based on generalized Hermite functions with weight . We also establish some basic results on generalized Hermite orthogonal approximations, which play an important role in spectral methods. As examples, the generalized Ginzburg-Landau equation in a population problem and an elliptic equation with a harmonic potential are considered. Related spectral schemes are proposed, and their convergence is proved. Numerical results demonstrate the spectral accuracy of this approach.
同济大学学报, 38(2010), 1496－1500.
-optimal designs for a hierarchically ordered system of regression models
Rongxian Yue and Xin Liu
Comp. Stat. Data Anal., 54(2010), 3458–3465.
-optimal designs are described for a kind of hierarchically ordered system of regression models with an r-dimensional response variable y. The components of y may be correlated with a known variance_covariance matrix . The present results show that –optimal designs for this system of regression models do not depend on . The –optimal designs are given for the systems of trigonometric and Haar wavelet regression models, respectively.
Bayesian robust designs for linear models with possible bias and correlated errors
Rongxian Yue and Xiaodong Zhou
Metrika 71 (2010), 1–15.
Consider the design problem for the approximately linear model with serially correlated errors. The correlated structure is the qth degree moving average process, MA(q), especially for q = 1, 2. The optimal design is derived by using Bayesian approach. The Bayesian designs derived with various priors are compared with the classical designs with respect to some specific correlated structures. The results show that any prior knowledge about the sign of the MA(q) process parameters leads to designs that are considerately more efficient than the classical ones based on homoscedastic assumptions.
Robust designs for models with possible bias and correlated errors
Rongxian Yue and Xiaodong Zhou
Appl. Math. J. Chin. Univ., 25(2010), 307-317.
This paper studies the model-robust design problem for general models with an unknown bias or contamination and the correlated errors. The true response function is assumed to be from a reproducing kernel Hilbert space and the errors are fitted by the q-th order moving average process MA(q), especially the MA(1) errors and the MA(2) errors. In both situations, design criteria are derived in terms of the average expected quadratic loss for the least squares estimation by using a minimax method. A case is studied and the orthogonality of the criteria is proved for this special response. The robustness of the design criteria is discussed through several numerical examples.
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