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

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SSRN

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

A diffuse-interface lattice Boltzmann method for fluid-particle interaction problems

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arXiv 2021年

作者： Liu, Jiao Huang, Changsheng Chaia, Zhenhua Shia, 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 Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, a diffuse-interface lattice Boltzmann method (DI-LBM) is developed for fluid-particle interaction problems. In this method, the sharp interface between the fluid and solid is replaced by a thin but nonzero thickness transition region named diffuse interface, where the physical variables varies continuously. In order to describe the diffuse interface, we introduce a smooth function, which is similar to the order parameter in phase-field model or the volume fraction of solid phase in the partially saturated lattice Boltzmann method (PS-LBM). In addition, to depict the fluid-particle interaction more accurately, a modified force term is also proposed and included in the evolution equation of the DI-LBM. Some classical problems are used to test the DI-LBM, and the results are in good agreement with some available theoretical and numerical works. Finally, it is also found that the DI-LBM is more efficient and accurate than the PS-LBM with the superposition model. © 2021, CC BY.

关键词： Kinetic theory

A structure-preserving numerical scheme for non-isothermal compressible flow in porous media: Energy conservation and entropy stability

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Physics of Fluids 2025年第5期37卷

作者： Kou, Jisheng Chen, Huangxin Sun, Shuyu Wang, Xiuhua School of Civil Engineering Shaoxing University Zhejiang Shaoxing312000 China School of Mathematics and Statistics Hubei Engineering University Hubei Xiaogan432000 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China School of Mathematical Sciences Tongji University Shanghai200092 China

Numerical simulation of non-isothermal flow in porous media has great applications in unconventional gas recovery. Reliable and promising numerical methods should be compatible with the first and second laws of thermodynamics. Toward this goal, by taking molar density and temperature as the primary unknowns, we propose an alternative formulation of the realistic non-isothermal gas flow model, which is proved to obey the first and second laws of thermodynamics. On the basis of this formulation, we propose an energy conservative and entropy stable numerical scheme;that is, the scheme can preserve the first law of thermodynamics (the energy conservation law) as well as the second law of thermodynamics (entropy stability). An important ingredient of the scheme is to derive a new discrete chemical potential utilizing the energy factorization approach, which can preserve the dissipation of original Helmholtz free energy. Some subtle temporal and spatial treatments have also been introduced to deal with the inherent coupling relationships between multiple variables. Using the local analysis method, we prove that the scheme can be compatible with both local and global forms of the first and second laws of thermodynamics. Numerical results validate our theoretical analysis and demonstrate conservative and stable features of the proposed scheme. © 2025 Author(s).

关键词： Entropy

Large deviations principle for stochastic delay differential equations with super-linearly growing coefficients

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arXiv 2022年

作者： Jin, Diancong Chen, Ziheng Zhou, Tau 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 Statistics Yunnan University Kunming650500 China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We utilize the weak convergence method to establish the Freidlin-Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the delay variable but also to the state variable. This work extends the existing results which develop the LDPs for SDDEs with super-linearly growing coefficients only with respect to the delay *** Codes 60H10, 60F10, 60H30 Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

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Acta Mathematica Scientia 2018年第2期38卷 651-672页

作者：张磊刘斌 School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton （gμHS2） system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative （gμHS2） system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative （gμHS2） system, we show that the semi-group of the solution operator {S（t）}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S（t）}t≥0 possesses a global attractor in the Sobolev space H2（S） × H2（S）.

关键词： Generalized two-component μ-Hunter-Saxton system viscous weakly dissipative existence global attractor period boundary conditions

Upper capacity entropy and packing entropy of saturated sets for amenable group actions

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arXiv 2021年

作者： Ren, Xiankun Zhang, Wenda Zhang, Yiwei College of Mathematics and Statistics Chongqing University Chongqing 401331 China College of Mathematics and Statistics Chongqing Jiaotong University Chongqing 400074 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

Let (X, G) be a G-action topological system, where G is a countable infinite discrete amenable group and X a compact metric space. In this paper we study the upper capacity entropy and packing entropy for systems with weaker version of specification. We prove that the upper capacity always carries full entropy while there is a variational principle for packing entropy of saturated sets. © 2021, CC BY-NC-ND.

关键词： Specifications

Data-driven method to learn the most probable transition pathway and stochastic differential equations

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arXiv 2021年

作者： Chen, Xiaoli Duan, Jinqiao Hu, Jianyu Li, Dongfang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Department of Mathematics National University of Singapore 119077 Singapore Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics College of Computing Illinois Institute of Technology ChicagoIL60616 United States

Transition phenomena between metastable states play an important role in complex systems due to noisy fluctuations. We introduce the Onsager-Machlup theory and the Freidlin-Wentzell theory to quantify rare events in stochastic differential equations. By the variational principle, the most probable transition pathway is the minimizer of the action functional, which is governed by the Euler-Lagrange equation. In this paper, the physics-informed neural networks (PINNs) are presented to compute the most probable transition pathway through computing the Euler-Lagrange equation. The convergence result for the empirical loss is obtained for the forward problem. Then we investigate the inverse problem to extract the stochastic differential equation from the most probable transition pathway data. Finally, several numerical experiments are presented to verify the effectiveness of our methods. Copyright © 2021, The Authors. All rights reserved.

关键词： Inverse problems

Stabilized multiple-relaxation implicit-explicit Runge-Kutta methods for nonlinear Gross-Pitaevskii equations

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Journal of Computational Physics 2025年

作者： Dongfang Li Xiaoxi Li Jiwei Zhang Zhimin Zhang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Statistics Wuhan University Wuhan 430072 China Hubei Key Laboratory of Computational Science Wuhan University Wuhan 430072 China Department of Mathematics Wayne State University Detroit 48202 MI USA

A novel class of stabilized multiple-relaxation implicit-explicit Runge–Kutta methods is proposed for solving nonlinear Gross–Pitaevskii equations. The methods conserve the mass and the energy simultaneously in the discrete sense, and achieve arbitrarily high-order accuracy in the temporal direction. Additionally, there are several highlights in this study. Firstly, this is the first attempt to apply the stabilization approach to develop relaxation-type schemes, which effectively avoids the blow-up of the relaxation parameters due to the nonlinearity and stiffness of the problems. Secondly, the governing equations that determine the relaxation parameters are newly designed to conserve the original invariants, which can be extended to conserve more invariants of other physical models. Thirdly, the schemes are linearly implicit after obtaining the relaxation parameters, and thus are easy to implement. Several numerical experiments are carried out to confirm the effectiveness and theoretical results of the proposed methods.

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Phase-field-based lattice Boltzmann model for immiscible incompressible N-phase flows

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Physical Review E 2020年第6期101卷 063310-063310页

作者： Xiaolei Yuan Hong Liang Zhenhua Chai 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 Department of Physics Hangzhou Dianzi University Hangzhou 310018 China

In this paper, we develop an efficient and alternative lattice Boltzmann (LB) model for simulating immiscible incompressible N-phase flows (N≥2) based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the N-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, (N−1) LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term. The developed model is first validated by two classical benchmark problems, including the tests of static droplets and the spreading of a liquid lens, the simulation results show that the current LB model is accurate and efficient for simulating incompressible N-phase fluid flows. To further demonstrate the capability of the LB model, two numerical simulations, including dynamics of droplet collision for four fluid phases and dynamics of droplets and interfaces for five fluid phases, are performed to test the developed model. The results show that the present model can successfully handle complex interactions among N (N≥2) immiscible incompressible flows.

关键词： Multiphase flows Lattice-Boltzmann methods