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  • Open Access


    A New Insight into the Differential Quadrature Method in Solving 2-D Elliptic PDEs

    Ying-Hsiu Shen1, Chein-Shan Liu1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.2, pp. 157-178, 2011, DOI:10.3970/cmes.2011.071.157

    Abstract When the local differential quadrature (LDQ) has been successfully applied to solve two-dimensional problems, the global method of DQ still has a problem by requiring to solve the inversions of ill-posed matrices. Previously, when one uses (n-1)th order polynomial test functions to determine the weighting coefficients with n grid points, the resultant n ×n Vandermonde matrix is highly ill-conditioned and its inversion is hard to solve. Now we use (m-1)th order polynomial test functions by n grid points that the size of Vandermonde matrix is m×n, of which m is much less than n. We More >

  • Open Access


    Numerical Simulations for Coupled Pair of Diffusion Equations by MLPG Method

    S. Abbasbandy1,2, V. Sladek3, A. Shirzadi1, J. Sladek3

    CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.1, pp. 15-38, 2011, DOI:10.3970/cmes.2011.071.015

    Abstract This paper deals with the development of a new method for solution of initial-boundary value problems governed by a couple of nonlinear diffusion equations occurring in the theory of self-organization in non-equilibrium systems. The time dependence is treated by linear interpolation using the finite difference method and the semi-discrete partial differential equations are considered in a weak sense by using the local integral equation method with approximating 2-d spatial variations of the field variables by the Moving Least Squares. The evaluation techniques are discussed and the applicability of the presented method is demonstrated on two More >

  • Open Access


    An Enhanced Fictitious Time Integration Method for Non-Linear Algebraic Equations With Multiple Solutions: Boundary Layer, Boundary Value and Eigenvalue Problems

    Chein-Shan Liu1, Weichung Yeih2, Satya N. Atluri3

    CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.3, pp. 301-324, 2010, DOI:10.3970/cmes.2010.059.301

    Abstract When problems in engineering and science are discretized, algebraic equations appear naturally. In a recent paper by Liu and Atluri, non-linear algebraic equations (NAEs) were transformed into a nonlinear system of ODEs, which were then integrated by a method labelled as the Fictitious Time Integration Method (FTIM). In this paper, the FTIM is enhanced, by using the concept of arepellorin the theory ofnonlinear dynamical systems, to situations where multiple-solutions exist. We label this enhanced method as MSFTIM. MSFTIM is applied and illustrated in this paper through solving boundary-layer problems, boundary-value problems, and eigenvalue problems with More >

  • Open Access


    A Fractional Order HIV Internal Viral Dynamics Model

    Caibin Zeng1, Qigui Yang1

    CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.1, pp. 65-78, 2010, DOI:10.3970/cmes.2010.059.065

    Abstract In this paper, a fractional order model is established to describe HIV internal viral dynamics involving HAART effect. First, the model is proved to possess non-negative solutions as desired in any population dynamics. Then, a detailed analysis is carried out to study the stability of equilibrium points. Numerical simulations are presented to illustrate the stability analysis. More >

  • Open Access


    New Interpretation to Variational Iteration Method: Convolution Iteration Method Based on Duhamel's Principle for Dynamic System Analysis

    Yunhua Li1,2, Yunze Li3, Chieh-Li Chen4, Cha’o-Kuang Chen5

    CMES-Computer Modeling in Engineering & Sciences, Vol.58, No.1, pp. 1-14, 2010, DOI:10.3970/cmes.2010.058.001

    Abstract Addressing the identification problem of the general Lagrange multiplier in the He's variational iteration method, this paper proposes a new kind of method based on Duhamel's principle for the dynamic system response analysis. In this method, we have constructed an analytical iteration formula in terms of the convolution for the residual error at the nth iteration, and have given a new interpretation to He's variational iteration method. The analysis illustrates that the computational result of this method is equal to that of He's variational iteration method on the assumption of considering the impulse response of More >

  • Open Access


    An Advanced Implicit Meshless Approach for the Non-linear Anomalous Subdiffusion Equation

    Y. T. Gu1, P. Zhuang2, F. Liu3

    CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.3, pp. 303-334, 2010, DOI:10.3970/cmes.2010.056.303

    Abstract Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation(FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by… More >

  • Open Access


    Fictitious Time Integration Method of Fundamental Solutions with Chebyshev Polynomials for Solving Poisson-type Nonlinear PDEs

    Chia-Cheng Tsai1, Chein-Shan Liu2, Wei-Chung Yeih3

    CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.2, pp. 131-152, 2010, DOI:10.3970/cmes.2010.056.131

    Abstract The fictitious time integration method (FTIM) previously developed by Liu and Atluri (2008a) is combined with the method of fundamental solutions and the Chebyshev polynomials to solve Poisson-type nonlinear PDEs. The method of fundamental solutions with Chebyshev polynomials (MFS-CP) is an exponentially-convergent meshless numerical method which is able to solving nonhomogeneous partial differential equations if the fundamental solution and the analytical particular solutions of the considered operator are known. In this study, the MFS-CP is extended to solve Poisson-type nonlinear PDEs by using the FTIM. In the solution procedure, the FTIM is introduced to convert More >

  • Open Access


    A High-Order Time and Space Formulation of the Unsplit Perfectly Matched Layer for the Seismic Wave Equation Using Auxiliary Differential Equations (ADE-PML)

    R. Martin1, D. Komatitsch1,2, S. D. Gedney3, E. Bruthiaux1,4

    CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.1, pp. 17-42, 2010, DOI:10.3970/cmes.2010.056.017

    Abstract Unsplit convolutional perfectly matched layers (CPML) for the velocity and stress formulation of the seismic wave equation are classically computed based on a second-order finite-difference time scheme. However it is often of interest to increase the order of the time-stepping scheme in order to increase the accuracy of the algorithm. This is important for instance in the case of very long simulations. We study how to define and implement a new unsplit non-convolutional PML called the Auxiliary Differential Equation PML (ADE-PML), based on a high-order Runge-Kutta time-stepping scheme and optimized at grazing incidence. We demonstrate More >

  • Open Access


    A Scalar Homotopy Method for Solving an Over/Under-Determined System of Non-Linear Algebraic Equations

    Chein-Shan Liu1, Weichung Yeih2, Chung-Lun Kuo3, Satya N. Atluri4

    CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.1, pp. 47-72, 2009, DOI:10.3970/cmes.2009.053.047

    Abstract Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): Fi(xj) = 0, i, j = 1,... ,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of algorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrix ∂Fi/∂xj and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration procedure for n-variables… More >

  • Open Access


    Solution Methods for Nonsymmetric Linear Systems with Large off-Diagonal Elements and Discontinuous Coefficients

    Dan Gordon1, Rachel Gordon2

    CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.1, pp. 23-46, 2009, DOI:10.3970/cmes.2009.053.023

    Abstract Linear systems with very large off-diagonal elements and discontinuous coefficients (LODC systems) arise in some modeling cases, such as those involving heterogeneous media. Such problems are usually solved by domain decomposition methods, but these can be difficult to implement on unstructured grids or when the boundaries between subdomains have a complicated geometry. Gordon and Gordon have shown that Björck and Elfving's (sequential) CGMN algorithm and their own block-parallel CARP-CG are very robust and efficient on strongly convection dominated cases (but without discontinuous coefficients). They have also shown that scaling the equations by dividing each equation… More >

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