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A phase-field-based lattice Boltzmann method for two-phase flows with the interfacial mass/heat transfer

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arXiv 2024年

作者： Chen, Baihui Zhan, Chengjie Chai, Zhenhua 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

In this work, we develop a phase-field-based lattice Boltzmann (LB) method for a two-scalar model of the two-phase flows with interfacial mass/heat transfer. Through the Chapman–Enskog analysis, we show that the present LB method can correctly recover the governing equations for phase field, flow field and concentration/temperature field. In particular, to derive the two-scalar equations for the mass/heat transfer, we propose a new LB model with an auxiliary source distribution function to describe the extra flux terms, and the discretizations of some derivative terms can be avoided. The accuracy and efficiency of the present method is also tested through several benchmark problems, and the influence of mass/heat transfer on the fluid viscosity is further considered by introducing an exponential relation. The numerical results show that the present LB method is suitable for the two-phase flows with interfacial mass/heat transfer. © 2024, CC BY.

关键词： Distribution functions

Regularized lattice Boltzmann method based maximum principle and energy stability preserving finite-difference scheme for the Allen-Cahn equation

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arXiv 2023年

作者： Chen, Ying Liu, Xi Chai, Zhenhua 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

The Allen-Cahn equation (ACE) inherently possesses two crucial properties: the maximum principle and the energy dissipation law. Preserving these two properties at the discrete level is also necessary in the numerical methods for the ACE. In this paper, unlike the traditionally top-down macroscopic numerical schemes which discretize the ACE directly, we first propose a novel bottom-up mesoscopic regularized lattice Boltzmann method based macroscopic numerical scheme for d (=1, 2, 3)-dimensional ACE, where the DdQ(2d+1) [(2d+1) discrete velocities in d-dimensional space] lattice structure is adopted. In particular, the proposed macroscopic numerical scheme has a second-order accuracy in space, and can also be viewed as an implicit-explicit finite-difference scheme for the ACE, in which the nonlinear term is discretized semi-implicitly, the temporal derivative and dissipation term of the ACE are discretized by using the explicit Euler method and second-order central difference method, respectively. Then we demonstrate that the proposed scheme can preserve the maximum bound principle and the original energy dissipation law at the discrete level under some conditions. Finally, some numerical experiments are conducted to validate our theoretical analysis. Copyright © 2023, The Authors. All rights reserved.

关键词： Maximum principle

Regularized Lattice Boltzmann Method Based Maximum Principle and Energy Stability Preserving Finite-Difference Scheme for the Allen-Cahn Equation

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SSRN

SSRN 2024年

关键词： Maximum principle

The Macroscopic Finite-Difference Scheme and Modified Equations of the General Propagation Multiple-Relaxation-Time Lattice Boltzmann Model

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SSRN

SSRN 2023年

作者： Chen, Ying Chai, Zhenhua Liu, Xi 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

In this paper, we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, two benchmark problems, Gauss hill problem and Poiseuille flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results are not only in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis. © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

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

Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues

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arXiv 2022年

作者： Chai, Zhenhua Yuan, Xiaolei Shi, Baochang 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 College of Mathematics and Information Science Hebei University Baoding071002 China

In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equation (NCDE) by extending our recent unified framework of MRT-LB method [Z.H. Chai and B.C. Shi, Phys. Rev. E 102, 023306 (2020)], where a rectangular equilibrium distribution function (REDF) [J.H. Lu et al, Phil. Trans. R. Soc. A 369, 2311-2319 (2011)] on a rectangular lattice is utilized. Due to the anisotropy of discrete velocities on a rectangular lattice, the third-order moment of REDF is inconsistent with that of popular LB method, and thus the previous unified framework of MRT-LB method cannot be directly applied to the NSEs using the REDF on the rectangular rDdQq (q discrete velocities in d-dimensional space, d ≥ 1) lattice. The macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method by properly selecting the relaxation sub-matrix S2 which is related to kinetic viscosity and bulk viscosity. While the rectangular lattice does not lead to the change of the zero-th, first and second-order moments of REDF, thus the unified framework of MRT-LB method can be directly applied to the NCDE. It should be noted that the RMRT-LB model for NSEs can be derived on the rDdQq lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of RMRT-LB method. Thanks to the block-lower triangular-relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existed) of the previous MRT-LB models can be obtained, including those based on raw (natural)-moment, central-moment, Hermite-moment, and central Hermite-moment, respectively. It is also found that when the standard or regular lattice is used, and the sound speed is taken as an adjustable parameter, the present RMRT-LB method becomes a new class of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB

关键词： Navier Stokes equations

Dvelopment and application of a fully integrated model for unsaturated-saturated nitrogen reactive transport

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Agricultural Water Management 2017年 180卷 35-49页

作者： Zhu, Yan Yang, Jinzhong Ye, Ming Sun, Huaiwei Shi, Liangsheng State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University WuhanHubei430072 China Department of Scientific Computing Florida State University TallahasseeFL32306 United States School of Hydropower and Information Engineering Huazhong University of Science and Technology Wuhan430074 China

This paper presents a fully integrated model to simulate coupled unsaturated-saturated flow and reactive transport of ammonium and nitrate, the two major nitrogen species. Based on our previous work of developing a coupled model of unsaturated-saturated flow and solute transport of a single species, the key contribution of this study is to develop a new mathematical model and the new computer code for a comprehensive list of biogeochemical reactions, which are necessary for simulating nitrogen reactive transport. The computer code is verified by comparing its simulated results with those obtained using another model for a synthetic case. Model calibration and validation are conducted for two real-world cases to simulate nitrogen reactive transport. The two cases are at the experimental plot scale with the sewage water irrigation and at the field scale with the septic tank effluent infiltration, respectively. Quantitative evaluation of the numerical modeling results indicates that the new model and computer code can accurately simulate field observations of moisture content, water table elevation, and nitrogen concentration. For example, the root mean square error for simulating moisture content can be as small as 0.01. The code is computationally efficient to capture spatial and temporal trends of nitrogen concentrations at the field scale with a large number of computational nodes and time steps. The fully integrated model is a promising tool for large-scale modeling of water flow and nitrogen transformation and transport under complex conditions in agricultural and urban environments. © 2016 Elsevier B.V.

关键词： Mean square error

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

Convergence analysis of a finite difference method for stochastic Cahn-Hilliard equation

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arXiv 2022年

作者： Hong, Jialin Jin, Diancong Sheng, Derui LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn–Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order 1. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in L1(R) to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly 83, where a difficulty we overcome is to derive the optimal Hölder continuity of the spatial semi-discrete numerical solution. Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems