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arXiv 2023年

作者： Chen, Ying 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 first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the equivalent macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain an equivalent macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis of the finite-difference scheme and the mesoscopic MRT-LB model. It is found that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis. Copyright © 2023, The Authors. All rights reserved.

关键词： Finite difference method

A consistent and conservative diffuse-domain lattice Boltzmann method for multiphase flows in complex geometries

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arXiv 2023年

作者： Liu, Xi Zhan, Chengjie Chen, Ying 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

modeling and simulation of multiphase flows in complex geomerties are challenging due to the complexity in describing the interface topology changes among different phases and the difficulty in implementing the boundary conditions on the irregular solid surface. In this work, we first developed a diffuse-domain (DD) based phase-field model for multiphase flows in complex geometries. In this model, the irregular fluid region is embedded into a larger and regular domain by introducing a smooth characteristic function. Then, the reduction-consistent and conservative phase-field equation for the multiphase field and the consistent and conservative Navier-Stokes equations for the flow field are reformulated as the diffuse-domain based consistent and conservative (DD-CC) equations where some additional source terms are added to reflect the effects of boundary conditions. In this case, there is no need to directly treat the complex boundary conditions on the irregular solid surface, and additionally, based on a matched asymptotic analysis, it is also shown that the DD-CC equations can converge to the original governing equations as the interface width parameter tends to zero. Furthermore, to solve the DD-CC equations, we proposed a novel and simple lattice Boltzmann (LB) method with a Hermite-moment-based collision matrix which can not only keep consistent and conservation properties, but also improve the numerical stability with a flexible parameter. With the help of the direct Taylor expansion, the macroscopic DD-CC equations can be recovered correctly from the present LB method. Finally, to test the capacity of LB method, several benchmarks and complex problems are considered, and the numerical results show that the present LB method is accurate and efficient for the multiphase flows in complex geomerties. © 2023, CC BY.

关键词： Boundary conditions

A Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann model for the coupled Burgers' equations

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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

In this work, a Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for d-dimensional coupled Burgers’ equations is developed. We first adopt the Cole-Hopf transformation where an intermediate variable θ is introduced to eliminate the nonlinear convection terms in the Burgers’ equations on the velocity u = (u1, u2, . . ., ud). In this case, a diffusion equation on the variable θ can be obtained, and particularly, the velocity u in the coupled Burgers’ equations is determined by the variable θ and its gradient term ∇θ. Then we develop the MRT-LB model with the natural moments for the d-dimensional transformed diffusion equation and present the corresponding macroscopic finite-difference scheme. At the diffusive scaling, the fourth-order modified equation of the developed MRT-LB model is derived through the Maxwell iteration method. With the aid of the free parameters in the MRT-LB model, we find that not only the consistent fourth-order modified equation can be obtained, but also the gradient term ∇θ with a fourth-order accuracy can be determined by the non-equilibrium distribution function, this indicates that theoretically, the MRT-LB model for d-dimensional coupled Burgers’ equations can achieve the fourth-order accuracy in space. Finally, some simulations are conducted to test the MRT-LB model, and the numerical results show that the proposed MRT-LB model has a fourth-order convergence rate, which is consistent with our theoretical analysis. Copyright © 2023, The Authors. All rights reserved.

关键词： Distribution functions

A note on the run length function for intermittent maps

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arXiv 2018年

作者： Cui, Hongfei Fang, Lulu Zhang, Yiwei Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences Wuhan430071 China Sun Yat-sen university Guangzhou510275 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

We study the run length function for intermittent maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittent maps. Our results are compensational to the Erd?os-Rényi law obtained by Denker and Nicol in [8]. Copyright © 2018, The Authors. All rights reserved.

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A General Fourth-Order Mesoscopic Multiple-Relaxation-Time Lattice Boltzmann Model and Equivalent Macroscopic Finite-Difference Scheme for Two-Dimensional Diffusion Equations

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SSRN

SSRN 2023年

关键词： Finite difference method

A rigorous computer aided estimation for gelfond exponent of weighted thue-morse sequences

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arXiv 2018年

作者： ZHANG, YIWEI YIN, KE WU, WANQUAN School of Mathematics and Statistics Center for Mathematical Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology 1037 Luoyu Road Wuhan430074 China School of Computer Science and Technology Huazhong University of Science and Technology 1037 Luoyu Road Wuhan430074 China

In this paper, we will provide a mathematically rigorous computer aided estimation for the exact values and robustness for Gelfond exponent of weighted Thue-Morse sequences. This result improves previous discussions on Gelfond exponent by Gelfond, Devenport, Mauduit, Rivat, Sarközy and Fan et. al.37C40, 90C27 Copyright © 2018, The Authors. All rights reserved.

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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

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

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