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Two classes of implicit–explicit multistep methods for nonlinear stiff initial-value problems

Applied Mathematics and computation 2014年 247卷 47-60页

作者： Aiguo Xiao Gengen Zhang Xing Yi School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China

The initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications. In order to reduce the computation cost, implicit–explicit (IMEX) methods are often applied to these problems, i.e. the stiff and non-stiff terms are discretized by using implicit and explicit methods, respectively. In this paper, we mainly consider the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, and present two classes of the IMEX multistep methods by combining implicit one-leg methods with explicit linear multistep methods and explicit one-leg methods, respectively. The order conditions and the convergence results of these methods are obtained. Some efficient methods are constructed. Some numerical examples are given to verify the validity of the obtained theoretical results.

关键词： Stiff problems Singular perturbation problems Implicit–explicit multistep methods One-leg methods Linear multistep methods Convergence

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Parallelization of a DEM code based on CPU-GPU heterogeneous architecture

Parallelization of a DEM code based on CPU-GPU heterogeneous...

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26th International Conference on Parallel computational Fluid Dynamics, ParCFD 2013

作者： Yue, Xiaoqiang Zhang, Hao Luo, Congshu Shu, Shi Feng, Chunsheng School of Mathematics and Computational Science Xiangtan University 411105 Hunan China Heat and Mass Transfer Technological Center Technical University of Catalonia 08222 Terrassa Spain Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University 411105 Hunan China

ISBN: (纸本)9783642539619

Particulate flows are commonly encountered in both engineering and environmental applications. The discrete element method (DEM) has attracted plentiful attentions since it can predict the whole motion of the particulate flow by monitoring every single particle. However the computational capability of the method relies strongly on the numerical scheme as well as the hardware environment. In this study, a parallelization of a DEM based code titled Trubal was implemented. Numerical simulations were carried out to show the benefits of this research. It is shown that the final parallel code gave a substantial acceleration on the Trubal. By simulating 6,000 particles using a NVIDIA Tesla C2050 card together with Intel Core-Dual 2.93 GHz CPU, an average speedup of 4.69 in computational time was obtained. © Springer-Verlag Berlin Heidelberg 2014.

关键词： Finite difference method

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Time-splitting finite difference method with the wavelet-adaptive grids for semiclassical Gross–Pitaevskii equation in supercritical case

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Journal of computational Physics 2014年 267卷 146-161页

作者： Xueyang Li Aiguo Xiao School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 PR China

The Gross–Pitaevskii equation is the model equation of the single-particle wave function in a Bose–Einstein condensation. A computation difficulty of the Gross–Pitaevskii equation comes from the semiclassical problem in supercritical case. In this paper, we apply a diffeomorphism to transform the original one-dimensional Gross–Pitaevskii equation into a modified equation. The adaptive grids are constructed through the interpolating wavelet method. Then, we use the time-splitting finite difference method with the wavelet-adaptive grids to solve the modified Gross–Pitaevskii equation, where the approximation to the second-order derivative is given by the Lagrange interpolation method. At last, the numerical results are given. It is shown that the obtained time-splitting finite difference method with the wavelet-adaptive grids is very efficient for solving the one-dimensional semiclassical Gross–Pitaevskii equation in supercritical case and it is suitable to deal with the local high oscillation of the solution.

关键词： Gross–Pitaevskii equation Wavelet-adaptive grids Time-splitting finite difference method Interpolating wavelet Lagrange interpolation method

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Jacobi spectral Galerkin methods for fractional integro-differential equations

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Calcolo 2014年第4期52卷 519-542页

作者： Yang, Yin Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan People’s Republic of China

We propose general spectral and pseudo-spectral Jacobi–Galerkin methods for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide rigorous error analysis for spectral and pseudo-spectral Jacobi–Galerkin methods, which show that the errors of the approximate solution decay exponentially in $$L^\infty $$ norm and weighted $$L^2$$ -norm. The numerical examples are given to illustrate the theoretical results.

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CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

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Acta Mathematica Scientia 2014年第3期34卷 673-690页

作者： Yang Yin Chen Yanping Huang Yunqing Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 School of Mathematical Sciences South China Normal University Guangzhou 510631

We propose and analyze a spectral Jacobi-collocation approximation for frac- tional order integro-differential equations of Volterra type. The fractional derivative is de- scribed in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L~∞ norm and weighted L~2-norm. The numerical examples are given to illustrate the theoretical results.

关键词： Spectral Jacobi-collocation method fractional order integro-differential equa- tions Caputo derivative

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A high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm

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Applied Mathematics and computation 2014年 227卷 212-221页

作者： Cunyun Nie Shi Shu Haiyuan Yu Qianjiang An Department of Mathematics and Physics The Hunan Institution of Engineering Xiangtan City Hunan 411104 China Hunan Key Laboratory for Computation & Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan University Hunan 411105 China Department of Mathematics The Institute of Tongren Guizhou 554300 China

The elliptic problem with nonlocal boundary condition is widely applied in the field of science and engineering. Firstly, we construct a linear finite element scheme for the nonlocal boundary problem, and derive the optimal L 2 error estimate. Then, based on the quadratic finite element and the extrapolation linear finite element methods, we present a composite scheme, and prove that it is convergent order three. Furthermore, we design an upper triangular preconditioning algorithm for the linear finite element discrete system. Finally, numerical results not only validate that the new algorithm is efficient, but also show that the new scheme is convergent order three, furthermore order four on uniform grids.

关键词： The elliptic problem with nonlocal boundary condition The finite element method An upper triangular preconditioner Algebraic multigrid method

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Multifractal analysis of weighted networks by a modified sandbox algorithm

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Scientific reports 2015年第1期5卷 17628页

作者： Yu-Qin Song Jin-Long Liu Zu-Guo Yu Bao-Gen Li Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China. College of Science Hunan University of technology Zhuzhou Hunan 412007 China. School of Mathematical Sciences Queensland University of Technology Brisbane Q4001 Australia.

Complex networks have attracted growing attention in many fields. As a generalization of fractal analysis, multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. Some algorithms for MFA of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this paper, a modified SB algorithm (we call it SBw algorithm) is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimensions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks - collaboration networks. It is found that the multifractality exists in these weighted networks, and is affected by their edge-weights.

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Laplacian normalization and random walk on heterogeneous networks for disease-gene prioritization

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computational biology and chemistry 2015年 57卷 21-8页

作者： Zhi-Qin Zhao Guo-Sheng Han Zu-Guo Yu Jinyan Li Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China. Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia. Electronic address: yuzg1970@***. Advanced Analytics Institute & Centre for Health Technologies University of Technology Sydney Broadway NSW 2007 Australia. Electronic address: jinyan.li@uts.edu.au.

Random walk on heterogeneous networks is a recently emerging approach to effective disease gene prioritization. Laplacian normalization is a technique capable of normalizing the weight of edges in a network. We use this technique to normalize the gene matrix and the phenotype matrix before the construction of the heterogeneous network, and also use this idea to define the transition matrices of the heterogeneous network. Our method has remarkably better performance than the existing methods for recovering known gene-phenotype relationships. The Shannon information entropy of the distribution of the transition probabilities in our networks is found to be smaller than the networks constructed by the existing methods, implying that a higher number of top-ranked genes can be verified as disease genes. In fact, the most probable gene-phenotype relationships ranked within top 3 or top 5 in our gene lists can be confirmed by the OMIM database for many cases. Our algorithms have shown remarkably superior performance over the state-of-the-art algorithms for recovering gene-phenotype relationships. All Matlab codes can be available upon email request.

关键词： Disease genes and phenotypes Heterogeneous network Laplacian normalization Leave-one-out cross-validation Random walk with restart Shannon information entropy

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Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

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Physical Review E 2014年第3期89卷 032814-032814页

作者： Jin-Long Liu Zu-Guo Yu Vo Anh Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2−H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2)〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic

关键词： TOPOLOGICAL property FRACTAL analysis BROWNIAN motion CORRELATION (Statistics) RECURSIVE sequences (Mathematics) COMPUTER networks

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Adaptive tetrahedral mesh generation by constrained centroidal voronoi-delaunay tessellations for finite element methods

Adaptive tetrahedral mesh generation by constrained centroid...

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作者： Chen, Jie Huang, Yunqing Wang, Desheng Xie, Xiaoping School of Mathematics and Statistics Xi'An Jiaotong University Xi'an 710049 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411005 China Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Singapore 637616 Singapore School of Mathematics Sichuan University Chengdu 610064 China

This article presents a tetrahedral mesh adaptivity algorithm for three-dimensional elliptic partial differential equations (PDEs) using finite element methods. The main issues involved are the mesh size and mesh quality, which have great influence on the accuracy of the numerical solution and computational cost. The first issue is addressed by a posteriori error estimator based on superconvergent gradient recovery. The second issue is solved by constrained centroidal Voronoi-Delaunay tessellations (CCVDT), which guarantees good quality tetrahedrons over a large class of mesh domains even, if the grid size varies a lot at any particular refinement level. The CCVDT enjoys the energy equidistribution property so that the errors are very well equidistributed with properly chosen sizing field (density function). And with this good property, a new refinement criteria is raised which is different from the traditional bisection refinement. © 2013 Wiley Periodicals, Inc.

关键词： Finite element method

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