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Precise asymptotics on the Birkhoff sums for dynamical systems

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arXiv 2020年

作者： Fang, Lulu Takahasi, Hiroki Zhang, Yiwei School of Science Nanjing University of Science and Technology Nanjing210094 China Department of Mathematics Keio University Yokohama223-8522 Japan School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).MSC Codes 37A44, 37A50, 60F10 © 2020, CC BY.

关键词： Dynamical systems

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state SN transport equation

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arXiv 2021年

作者： Ren, Yupeng Xing, Yulong Wang, Dean Qiu, Jianxian School of Mathematical Sciences Xiamen University XiamenFujian361005 China Department of Mathematics Ohio State University ColumbusOH43210 United States Department of Mechanical and Aerospace Engineering Ohio State University ColumbusOH43210 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian361005 China

In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state SN transport equation in the finite volume framework. It is well-known that the SN transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonstrate the asymptotic preserving property of the high order finite volume HWENO method, by showing that its cell-edge and cell-average fluxes possess the thick diffusion limit. Numerical results in both one- and two- dimensions are presented to validate its asymptotic preserving property. A hybrid strategy to compute the nonlinear weights in the HWENO reconstruction is introduced to save computational cost. Extensive one- and two-dimensional numerical experiments are performed to verify the accuracy, asymptotic preserving property and positivity of the proposed HWENO FSM. Copyright © 2021, The Authors. All rights reserved.

关键词： Diffusion

Efficient Numerical Computation of Time-Fractional Nonlinear Schrodinger Equations in Unbounded Domain

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Communications in Computational Physics 2019年第1期25卷 218-243页

作者： Jiwei Zhang Dongfang Li Xavier Antoine Beijing Computational Science Research Center Beijing 100193P.R.China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Institut Elie Cartan de Lorraine UMR CNRS 7502Universit́e de LorraineInria Nancy-Grand EstSPHINX TeamF-54506 Vandoeuvre-l`es-Nancy CedexFrance

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded *** first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization ***,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is *** accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional *** resulting algorithm is highly efficient for long time ***,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.

关键词： Time-fractional nonlinear Schrodinger equation absorbing boundary condition sta-bility analysis convergence analysis sum-of-exponentials approximation

A block triple-relaxation-Time lattice Boltzmann model for nonlinear anisotropic convection-diffusion equations

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arXiv 2019年

作者： Zhao, Yong Wu, Yao Chai, Zhenhua Shi, Baochang Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

A block triple-relaxation-Time (B-TriRT) lattice Boltzmannmodel for general nonlinear anisotropic convection-diffusion equations (NACDEs) is proposed, and the Chapman-Enskog analysis shows that the present B-TriRT model can recover the NACDEs correctly. There are some striking features of the present B-TriRT model: firstly, the relaxation matrix of B-TriRT model is partitioned into three relaxation parameter blocks, rather than a diagonal matrix in general multiple-relaxation-Time (MRT) model;secondly, based on the analysis of half-way bounce-back (HBB) scheme for Dirichlet boundary conditions, we obtain an expression to determine the relaxation parameters;thirdly, the anisotropic diffusion tensor can be recovered by the relaxation parameter block that corresponds to the first-order moment of nonequilibriumdistribution function. A number of simulations of isotropic and anisotropic convection-diffusion equations are conducted to validate the present B-TriRT model. The results indicate that the present model has a second-order accuracy in space, and is also more accurate and more stable than some available lattice Boltzmann models. Copyright © 2019, The Authors. All rights reserved.

关键词： Diffusion in liquids

Mixed bounce-back boundary scheme of the general propagation lattice Boltzmann method for advection-diffusion equations

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Physical Review E 2019年第6期99卷 063316-063316页

作者： Xiuya Guo Zhenhua Chai Shengyong Pang Yong Zhao Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China State Key Laboratory of Material Processing and Die & Mould Technology Huazhong University of Science and Technology Wuhan 430074 China

In this work, a mixed bounce-back boundary scheme of general propagation lattice Boltzmann (GPLB) model is proposed for isotropic advection-diffusion equations (ADEs) with Robin boundary condition, and a detailed asymptotic analysis is also conducted to show that the present boundary scheme for the straight walls has a second-order accuracy in space. In addition, several numerical examples, including the Helmholtz equation in a square domain, the diffusion equation with sinusoidal concentration gradient, one-dimensional transient ADE with Robin boundary and an ADE with a source term, are also considered. The results indicate that the numerical solutions agree well with the analytical ones, and the convergence rate is close to 2.0. Furthermore, through adjusting the two parameters in the GPLB model properly, the present boundary scheme can be more accurate than some existing lattice Boltzmann boundary schemes.

关键词： Lattice-Boltzmann methods

Maxwell-Stefan-theory-based lattice Boltzmann model for diffusion in multicomponent mixtures

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Physical Review E 2019年第2期99卷 023312-023312页

作者： Zhenhua Chai Xiuya Guo Lei Wang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China

The phenomena of diffusion in multicomponent (more than two components) mixtures are universal in both science and engineering, and from the mathematical point of view, they are usually described by the Maxwell-Stefan (MS)-theory-based diffusion equations where the molar average velocity is assumed to be zero. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures and also perform a Chapman-Enskog analysis to show that the MS continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS-theory-based diffusion equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model and find that the results are in good agreement with the previous work. Besides, the reverse diffusion, osmotic diffusion, and diffusion barrier phenomena are also captured. Finally, compared to the kinetic-theory-based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can still be implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model.

关键词： Lattice-Boltzmann methods

Discrete effects on some boundary schemes of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations

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arXiv 2019年

作者： Wu, Yao Zhao, Yong Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we perform a more general analysis on the discrete effects of some boundary schemes of the popular one- to three-dimensional DnQq multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation (CDE). Investigated boundary schemes include anti-bounce-back(ABB) boundary scheme, bounce-back(BB) boundary scheme and non-equilibrium extrapolation(NEE) boundary scheme. In the analysis, we adopt a transform matrix M constructed by natural moments in the evolution equation, and the result of ABB boundary scheme is consistent with the existing work of orthogonal matrix M. We also find that the discrete effect does not rely on the choice of transform matrix, and obtain a relation to determine some of the relaxation-time parameters which can be used to eliminate the numerical slip completely under some assumptions. In this relation, the weight coefficient is considered as an adjustable parameter which makes the parameter adjustment more flexible. The relaxation factors associated with second moments can be used to eliminate the numerical slip of ABB boundary scheme and BB boundary scheme while the numerical slip can not be eliminated of NEE boundary scheme. Furthermore, we extend the relations to complex-valued CDE, several numerical examples are used to test the relations. Copyright © 2019, The Authors. All rights reserved.

关键词： Diffusion in liquids

Generalized lattice boltzmann method: modeling, analysis, and elements

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arXiv 2019年

作者： Shi, Baochang Chai, Zhenhua School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we first present a unified framework for the modelling of generalized lattice Boltzmann method (GLBM). We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes equations and nonlinear convection-diffusion equations from the GLBM, and show that from mathematical point of view, these four analysis methods are equivalent to each other. Finally, we give some elements that are needed in the implementation of the GLBM, and also find that some available LB models can be obtained from this GLBM. Copyright © 2019, The Authors. All rights reserved.

关键词： Nonlinear equations

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

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arXiv 2020年

作者： Chen, Jianyu Zhang, Hong-Kun Zhang, Yiwei School of Mathematical Sciences Center for Dynamical Systems and Differential Equations Soochow University SuzhouJiangsu215006 China Department of Mathematics and Statistics University of Massachusetts Amherst AmherstMA01003 United States School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Hua-Zhong University of Sciences and Technology Wuhan430074 China

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernov’s one-step expansion at q-scale and eventually covers a magnet interval. Therefore, our approach is particularly powerful for maps whose inverse Jacobian has low regularity and those who does not satisfy the big image property. The main ingredients of our coupling method are two crucial lemmas: the growth lemma in terms of the characteristic Z function and the covering ratio lemma over the magnet interval. We first prove the existence of an absolutely continuous invariant measure. What is more important, we further show that the growth lemma enables the liftablity of the Lebesgue measure to the associated Hofbauer tower, and the resulting invariant measure on the tower admits a decomposition of Pesin-Sinai type. Furthermore, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem). For the first time, we are able to make a direct relation between the mixing rates and the Z function, see (2.7). The novelty of our results relies on establishing the regularity of invariant density, as well as verifying the stochastic properties for a large class of unbounded observables. Finally, we verify our assumptions for several well known examples that were previously studied in the literature, and unify results to these examples in our framework. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems