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Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations

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arXiv 2019年

作者： Shang, Jinlong Chai, Zhenhua Wang, Huili 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 School of Mathematics and Computer Science Wuhan Textile University Wuhan430073 China

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for general nonlinear convection-diffusion equation (NCDE), and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and find that the numerical results are in good agreement with analytical solutions and the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, DUGKS can also adopt the nonuniform mesh. Besides, we performed some comparisons among the DUGKS, finite-volume lattice Boltzmann model (FV-LBM), single-relaxation-time lattice Boltzmann model (SLBM) and multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the DUGKS model is more accurate than FV-LBM, more stable than SLBM, and almost has the same accuracy as the MRT-LBM. Besides, the using of non-uniform mesh may make DUGKS model more flexible. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite volume method

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

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Communications in Computational Physics 2018年第6期24卷 86-103页

作者： Dongfang Li Hong-Lin Liao Weiwei Sun Jilu Wang Jiwei Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Department of Mathematics City University of Hong KongKowloonHong Kong Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing211106China Department of Scientific Computing Florida StateUniversityTallahasseeFL 32306USA Beijing Computational Science Research Center Beijing 100094China.

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element *** analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type *** this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional *** terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear *** theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

关键词： Time-fractional nonlinear parabolic problems L1-Galerkin FEMs Error estimates discrete fractional Gronwall type inequality Linearized schemes

General propagation lattice Boltzmann model for nonlinear advection-diffusion equations

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Physical Review E 2018年第4期97卷 043310-043310页

作者： Xiuya Guo Baochang Shi Zhenhua Chai 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

In this paper, a general propagation lattice Boltzmann model is proposed for nonlinear advection-diffusion equations (NADEs), and the Chapman-Enskog analysis shows that the NADEs with variable coefficients can be recovered correctly from the present model. We also perform some simulations of the linear advection-diffusion equation, nonlinear heat conduction equation, NADEs with anisotropic diffusion, and variable coefficients to test the present model, and find that the numerical results agree well with the corresponding analytical solutions. Moreover, it is also shown that by properly adjusting the two free parameters introduced into the propagation step, the present model could be more stable and more accurate than the standard lattice Bhatnagar-Gross-Krook model.

关键词： Lattice-Boltzmann methods

Ergodic optimization theory for a class of typical maps

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arXiv 2019年

作者： Huang, Wen Lian, Zeng Ma, Xiao Xu, Leiye Zhang, Yiwei Wu Wen-Tsun Key Laboratory of Mathematics USTC Chinese Academy of Sciences Department of Mathematics University of Science and Technology of China Hefei Anhui China College of Mathematical Sciences Sichuan University Chengdu Sichuan610016 China 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

In this article, we consider the weighted ergodic optimization problem of a class of dynamical systems T: X → X where X is a compact metric space and T is Lipschitz continuous. We show that once T: X → X satisfies both the Anosov shadowing property (ASP) and the Mañé-Conze-Guivarc'h-Bousch property (MCGBP), the minimizing measures of generic Hölder observables are unique and supported on a periodic orbit. Moreover, if T: X → X is a subsystem of a dynamical system f: M → M (i.e. X ⊂ M and f|X= T ) where M is a compact smooth manifold, the above conclusion holds for C1observables. Note that a broad class of classical dynamical systems satisfies both ASP and MCGBP, which includes Axiom A attractors, Anosov diffeomorphisms and uniformly expanding maps. Therefore, the open problem proposed by Yuan and Hunt in [YH] for C1-observables is solved consequentially. Copyright © 2019, The Authors. All rights reserved.

关键词： Dynamical systems

Ergodic optimization theory for axiom a flows

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arXiv 2019年

作者： Huang, Wen Lian, Zeng Ma, Xiao Xu, Leiye Zhang, Yiwei Wu Wen-Tsun Key Laboratory of Mathematics USTC Chinese Academy of Sciences Department of Mathematics University of Science and Technology of China Hefei Anhui China College of Mathematical Sciences Sichuan University Chengdu Sichuan610016 China 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 article, we consider the weighted ergodic optimization problem Axiom A attractors of a C2flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space C0,α(α∈ (0, 1]) or C1the minimizing measure is unique and is supported on a periodic orbit. Copyright © 2019, The Authors. All rights reserved.

关键词： Site selection

Invariant foliations for stochastic dynamical systems with multiplicative stable Lévy noise

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arXiv 2018年

作者： Chao, Ying Wei, Pingyuan Yuan, Shenglan 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

This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Lévy noise. We first establish the existence of stable and unstable foliations for this kind of system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Phase retrieval for sub-Gaussian measurements

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arXiv 2019年

作者： Gao, Bing Liu, Haixia Wang, Yang School of Mathematical Sciences Nankai University Tianjin300071 China 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 Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong

Generally, phase retrieval problem can be viewed as the reconstruction of a function/signal from only the magnitude of the linear measurements. These measurements can be, for example, the Fourier transform of the density function. Computationally the phase retrieval problem is very challenging. Many algorithms for phase retrieval are based on i.i.d. Gaussian random measurements. However, Gaussian random measurements remain one of the very few classes of measurements. In this paper, we develop an efficient phase retrieval algorithm for sub-gaussian random frames. We provide a general condition for measurements and develop a modified spectral initialization. In the algorithm, we first obtain a good approximation of the solution through the initialization, and from there we use Wirtinger Flow to solve for the solution. We prove that the algorithm converges to the global minimizer linearly. Copyright © 2019, The Authors. All rights reserved.

关键词： Gaussian distribution

A unified framework for asymptotic analysis and computation of finite Hankel transform

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arXiv 2018年

作者： Wang, Haiyong School of Mathematics and Statistics Huazhong University of Science and Technology China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper we present a unified framework for asymptotic analysis and computation of the finite Hankel transform. This framework enables us to derive asymptotic expansions of the transform, including the cases where the oscillator has zeros and stationary points. As a consequence, two efficient and affordable methods for computing the transform numerically are developed and a detailed analysis of their asymptotic error estimate is carried out. Numerical examples are provided to confirm our analysis.65D30, 65D32, 65R10, 41A60 Copyright © 2018, The Authors. All rights reserved.

关键词： Asymptotic analysis

Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows

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Physical Review E 2018年第3期97卷 033309-033309页

作者： Hong Liang Jiangrong Xu Jiangxing Chen Huili Wang Zhenhua Chai Baochang Shi Department of Physics Hangzhou Dianzi University Hangzhou 310018 China 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

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. Lastly, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature.

关键词： Interfacial flows Kelvin-Helmholtz instability Multiphase flows Rayleigh-Taylor instability Lattice-Boltzmann methods