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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

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Most Probable Transition Pathways and Maximal Likely Trajectories in a Genetic Regulatory System

arXiv

引用

arXiv 2018年

作者： Cheng, Xiujun Wang, Hui Wang, Xiao Duan, Jinqiao Li, Xiaofan Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Statistics Henan University Kaifeng475000 China Department of Applied Mathematics Illinois Institute of Technology Chicago60616 United States School of Mathematics and Statistics Zhengzhou University Zhengzhou450001 China

We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator’s concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager-Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period);(ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise;(b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription. These findings provide insights for further experimental research, in order to achieve or to avoid sp

关键词： Trajectories

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Phase-field based lattice Boltzmann method for containerless freezing

arXiv

引用

arXiv 2024年

作者： Huang, Jiangxu Wang, Lei Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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, a phase-field based lattice Boltzmann (LB) method is proposed to simulate solid-liquid phase change phenomena in multiphase systems. The method couples the thermal properties of the solidification front with the dynamics of the droplet interface, which enables the description of the complex interfacial changes during solid-liquid phase change process. The volume expansion or shrinkage of the droplet caused by the change of density during phase change is represented by adding a mass source term to the continuum equation. We first test the developed LB method by the three-phase Stefan problem and the droplet solidification on a cold surface, and the numerical results are in good agreement with the analytical and experimental solutions. Then the LB method is used to study the solidification problem with bubbles. The results show that the model can accurately capture the effect of bubbles on the solidification process, which is in good agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle. Copyright © 2024, The Authors. All rights reserved.

关键词： Freezing

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ASYMPTOTIC ERROR DISTRIBUTION OF ACCELERATED EXPONENTIAL EULER METHOD FOR PARABOLIC SPDES

arXiv

引用

arXiv 2024年

作者： Hong, Jialin Jin, Diancong Wang, Xu Yang, Guanlin LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 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 China

The asymptotic error distribution of numerical methods applied to stochastic ordinary differential equations has been well studied, which characterizes the evolution pattern of the error distribution in the small step-size regime. It is still open for stochastic partial differential equations whether the normalized error process of numerical methods admits a nontrivial limit distribution. We answer this question by presenting the asymptotic error distribution of the temporal accelerated exponential Euler (AEE) method when applied to parabolic stochastic partial differential equations. In order to overcome the difficulty caused by the infinite-dimensional setting, we establish a uniform approximation theorem for convergence in distribution. Based on it, we derive the limit distribution of the normalized error process of the AEE method by studying the limit distribution of its certain appropriate finite-dimensional approximation process. As applications of our main result, the asymptotic error distribution of a fully discrete AEE method for the original equation and that of the AEE method for a stochastic ordinary differential equation are also obtained. Copyright © 2024, The Authors. All rights reserved.

关键词： Ordinary differential equations

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A Phase-Field-Based Multiple-Relaxation-Time Lattice Boltzmann Method for Incompressible Multiphase Flows with Density and Viscosity Contrasts

SSRN

引用

SSRN 2022年

作者： Yuan, Xiaolei Wu, Yao Zhang, Chunhua Chai, Zhenhua Shi, Baochang College of Mathematics and Information Science Hebei University Baoding071002 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Guangdong Shenzhen518055 China

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure $p$ is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures. In addition, theoretical analysis and modeling are both carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model and is very easy to analyze and implement. We then test the present model (mainly CLBM) by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio ($1000$). © 2022, The Authors. All rights reserved.

关键词： Multiphase flow

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A Two-Scale Reduced Model for Darcy Flow in Fractured Porous Media

引用

Procedia Computer Science 2016年 80卷 1324-1333页

作者： Huangxin Chen Shuyu Sun School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian 361005 China and King Abdullah University of Science and Technology (KAUST) Physical Science and Engineering Division (PSE) Thuwal 23955-6900 Saudi Arabia King Abdullah University of Science and Technology (KAUST) Physical Science and Engineering Division (PSE) Thuwal 23955-6900 Saudi Arabia

In this paper, we develop a two-scale reduced model for simulating the Darcy flow in two- dimensional porous media with conductive fractures. We apply the approach motivated by the embedded fracture model (EFM) to simulate the flow on the coarse scale, and the effect of fractures on each coarse scale grid cell intersecting with fractures is represented by the discrete fracture model (DFM) on the fine scale. In the DFM used on the fine scale, the matrix-fracture system are resolved on unstructured grid which represents the fractures accurately, while in the EFM used on the coarse scale, the flux interaction between fractures and matrix are dealt with as a source term, and the matrix-fracture system can be resolved on structured grid. The Raviart-Thomas mixed finite element methods are used for the solution of the coupled flows in the matrix and the fractures on both fine and coarse scales. Numerical results are presented to demonstrate the efficiency of the proposed model for simulation of flow in fractured porous media.

关键词： Fractured porous media two-scale discrete fracture model embedded fracture model mixed finite element method

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Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motion

arXiv

引用

arXiv 2021年

作者： Li, Min Hu, Yaozhong Huang, Chengming Wang, Xiong School of Mathematics and Physics Center for Mathematical Sciences China University of Geosciences Wuhan China Department of Mathematical and Statistical Sciences University of Alberta Edmonton Canada School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modelling and Scientific Computing Huazhong University of Science and Technology Wuhan China

In this paper, we study the mean square stability of the solution and its stochastic theta scheme for the following stochastic differential equations drive by fractional Brownian motion with Hurst parameter H ∈ (1/2,1): dX(t) = f(t,X(t))dt+g(t,X(t))dBH(t). Firstly, we consider the special case when f(t,X) = −λκtκ-1X and g(t,X) = µX. The solution is explicit and is mean square stable when κ ≥ 2H. It is proved that if the parameter 2H ≤ κ ≤ 3/2 and √3/2·e/√3/2·e+1 (≈ 0.77) ≤ θ ≤ 1 or κ > 3/2 and 1/2 MSC Codes 65C30, 65L20, 60G22, 60H10, 33C15 Copyright © 2021, The Authors. All rights reserved.

关键词： Stochastic systems

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The Exponential Transformation Based Lattice Boltzmann Model for Convection-Diffusion Equation

SSRN

引用

SSRN 2022年

作者： Zhang, Ting Cui, Shuqi Hong, Ning Shi, Baochang College of Sciences Wuhan University of Science and Technology Wuhan430081 China School of General Education Wuchang University of Technology Wuhan430223 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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, an exponential transformation based lattice Boltzmann (LB) model for solving the n -dimensional ( n D) convection-diffusion equation(CDE) is developed. Firstly, a class of exponential transformation is proposed to convert the n D CDE into a diffusion equation. Then, the converted diffusion equation is solved by the LB model. So, compared to the available LB models for CDE, the present LB model can eliminate the difficulty in treating the convection term. With the direct Taylor expansion method, it is shown that the CDE can be exactly derived from the exponential transformation based LB model. Finally, a variety of numerical tests have been conducted to validate the present LB model. It can be found that the numerical results agree well with the analytical solutions. Moreover, we also find that the present LB model has a second-order convergence rate in space, and it is more effective and more stable than the previous LB model for the CDE. © 2022, The Authors. All rights reserved.

关键词： Boltzmann equation

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Multiple-distribution-function finite-difference lattice Boltzmann method for incompressible Navier-Stokes equation

arXiv

引用

arXiv 2023年

作者： Chen, Xinmeng Chai, Zhenhua Zhao, Yong Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 Statistics Changsha University of Science and Technology Hunan Changsha410114 China

In this paper, a multiple-distribution-function finite-difference lattice Boltzmann method (MDF-FDLBM) is proposed for the convection-diffusion system based incompressible Navier-Stokes equations (NSEs). By Chapman Enskog analysis, the convection-diffusion system based incompressible NSEs can be recovered from MDF-FDLBM. Some quantities, including the velocity gradient, velocity divergence, strain rate tensor, shear stress and vorticity, can be computed locally by the first-order moment of the non-equilibrium distribution function. Through the von Neumann analysis, we conduct the stability analysis for the MDF-FDLBM and incompressible finite-difference lattice Boltzmann method (IFDLBM). It is found that the IFDLBM will be more stable than that of MDF-FDLBM with small kinematic viscosity, and the MDF-FDLBM will be more stable than that of IFDLBM with large Courant-Friedrichs-Lewy condition number. Finally, some simulations are conducted to validate the MDF-FDLBM. The results agree well with the analytical solutions and previous results. Through the numerical testing, we find that the MDF-FDLBM has a second-order convergence rate in space and time. The MDF-FDLBMcombined with non-uniform grid also works well. Meanwhile, compared with IFDLBM, it can be found that MDF-FDLBM offers higher accuracy and computational efficiency, reducing computation time by more than 36%. © 2023, CC BY.

关键词： Distribution functions

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Numerical Simulation of Power-Law Fluid Flow in a Trapezoidal Cavity using the Incompressible Finite-Difference Lattice Boltzmann Method

arXiv

引用

arXiv 2023年

In this paper, a numerical investigation of power-law fluid flow in the trapezoidal cavity has been conducted by incompressible finite-difference lattice Boltzmann method (IFDLBM). By designing the equilibrium distribution function, the Navier-Stokes equations (NSEs) can be recovered exactly. Through the coordinate transformation method, the body-fitted grid in physical region is transformed into a uniform grid in computational region. The effect of Reynolds (Re) number, the power-law index n and the vertical angle θ on the trapezoidal cavity are investigated. According to the numerical results, we come to some conclusions. For low Re number Re = 100, it can be found that the behavior of power-law fluid flow becomes more complicated with the increase of n. And as vertical angle θ decreases, the flow becomes smooth and the number of vortices decreases. For high Re numbers, the flow development becomes more complex, the number and strength of vortices increase. If the Reynolds number increases further, the power-law fluid will changes from steady flow to periodic flow and then to turbulent flow. For the steady flow, the lager the θ, the more complicated the vortices. And the critical Re number from steady to periodic state decreases with the decrease of power-law index n. © 2023, CC BY.

关键词： Reynolds number

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