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arXiv 2025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 The State Key Laboratory of Intelligent Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China

This paper discusses how to develop a high-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the general d(≥ 1)-dimensional diagonal-anisotropic diffusion equation. Such an MRT-LB model considers the transformation matrix constructed in a natural way and the DdQ(2d2 + 1) [(2d2 + 1) discrete velocities in d-dimensional space] lattice structure. A key step in developing the high-order MRT-LB model is to determine the additional adjustable relaxation parameters and weight coefficients, which are used to eliminate the truncation errors at some certain orders of the MRT-LB model, while ensuring the stability of the MRT-LB model. In this work, we first present a unified MRT-LB model for the d-dimensional diagonal-anisotropic diffusion equation. Then, through the direct Taylor expansion, we analyze the macroscopic modified equations of the MRT-LB model up to fourth-order at the diffusive scaling, and further derive the conditions that ensure the MRT-LB model to be fourth-order consistent with the diagonal-anisotropic diffusion equation. In particular, when the diagonal-anisotropic diffusion equation is reduced to the isotropic type, we propose another MRT-LB model with the DdQ(2d+ 1) lattice structure [fewer discrete velocities than the DdQ(2d2 + 1) lattice structure], and the fourth-order conditions are similarly derived. Additionally, we also construct the fourth-order initialization scheme for the present LB method. After that, the condition which guarantees that the MRT-LB model can satisfy the stability structure is explicitly given, and we would like to point out that from a numerical perspective, once the stability structure is satisfied, the MRT-LB model must be L2 stable. In combination with the fourth-order consistent and L2 stability conditions, the relaxation parameters and weight coefficients of the MRT-LB model can be automatically given by a simple computer code. Finally, we perform numerical simulations of several benchmark problems, and f

关键词： Anisotropy

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An efficient spectral method for the fractional Schrödinger equation on the real line

arXiv

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arXiv 2022年

作者： Shen, Mengxia Wang, Haiyong 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 fractional Schrödinger equation (FSE) on the real line arises in a broad range of physical settings and their numerical simulation is challenging due to the nonlocal nature and the power law decay of the solution at infinity. In this paper, we propose a new spectral discretization scheme for the FSE in space based upon Malmquist-Takenaka functions. We show that this new discretization scheme achieves much better performance than existing discretization schemes in the case where the underlying FSE involves the square root of the Laplacian, while in other cases it also exhibits comparable or even better performance. Numerical experiments are provided to illustrate the effectiveness of the proposed method. © 2022, CC BY.

关键词： Laplace transforms

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Improved phase-field-based lattice Boltzmann method for thermocapillary flow

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Physical Review E 2022年第1期105卷 015314-015314页

作者： Liqing Yue Zhenhua Chai Huili Wang 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 School of Mathematical and Computer Sciences Wuhan Textile University Wuhan 430074 China

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equations can be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.

关键词： Drops & bubbles Multiphase flows Lattice-Boltzmann methods Navier-Stokes equation

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EFFECT OF RANDOM NOISES ON PATHWISE SOLUTIONS TO THE HIGH-DIMENSIONAL MODIFIED EULER-POINCARÉ SYSTEM

arXiv

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arXiv 2021年

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

In this paper, we study the Cauchy problem for the stochastically perturbed high-dimensional modified Euler-Poincaré system (MEP2) on the torus Td, d ≥ 1. We first establish a local well-posedness framework in the sense of Hadamard for the MEP2 driven by general nonlinear multiplicative noises. Then two kinds of global existence and uniqueness results are demonstrated: One indicates that the MEP2 perturbed by nonlocal-type random noises with proper intensity admits a unique large global strong solution;The other one infers that, if the initial data is sufficiently small, then the MEP2 perturbed by linear multiplicative noise has a unique global solution with high probability. In the case of one dimension, we find that the stochastic MEP2 will break down in finite time when the initial data meets appropriate shape condition. Copyright © 2021, The Authors. All rights reserved.

关键词： Stochastic systems

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Phase-field modeling of dendritic growth with gas bubbles in the solidification of binary alloys

arXiv

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arXiv 2024年

作者： Zhan, Chengjie Chai, Zhenhua Sun, Dongke Shi, Baochang Geng, Shaoning Jiang, Ping 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 Mechanical Engineering Southeast University Nanjing211189 China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China

In this work, a phase-field model is developed for the dendritic growth with gas bubbles in the solidification of binary alloys. In this model, a total free energy for the complex gas-liquid-dendrite system is proposed through considering the interactions of gas bubbles, liquid melt and solid dendrites, and it can reduce to the energy for gas-liquid flows in the region far from the solid phase, while degenerate to the energy for thermosolutal dendritic growth when the gas bubble disappears. The governing equations are usually obtained by minimizing the total free energy, but here some modifications are made to improve the capacity of the conservative phase-field equation for gas bubbles and convection-diffusion equation for solute transfer. Additionally, through the asymptotic analysis of the thin-interface limit, the present general phase-field model for alloy solidification can match the corresponding free boundary problem, and it is identical to the commonly used models under a specific choice of model parameters. Furthermore, to describe the fluid flow, the incompressible Navier-Stokes equations are adopted in the entire domain including gas, liquid, and solid regions, where the fluid-structure interaction is considered by a simple diffuse-interface method. To test the present phase-field model, the lattice Boltzmann method is used to study several problems of gas-liquid flows, dendritic growth as well as the solidification in presence of gas bubbles, and a good performance of the present model for such complex problems is observed. © 2024, CC BY-NC-ND.

关键词： Solidification

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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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Physical Review E 2023年第1期108卷 015304-015304页

作者： Zhenhua Chai (柴振华) Xiaolei Yuan (袁晓垒) Baochang Shi (施保昌) School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 College of Mathematics and Information Science Hebei University Baoding 071002 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 the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)], where an equilibrium distribution function (EDF) [Lu et al., Philos. Trans. R. Soc. A 369, 2311 (2011)] on a rectangular lattice is utilized. The anisotropy of the lattice tensor on a rectangular lattice leads to anisotropy of the third-order moment of the EDF, which is inconsistent with the isotropy of the viscous stress tensor of the NSEs. To eliminate this inconsistency, we extend the relaxation matrix related to the dynamic and bulk viscosities. As a result, the macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method. Whereas the rectangular lattice does not lead to the change of the zero-, first- and second-order moments of the EDF, the unified framework of the 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 (q discrete velocities in d-dimensional space, d≥1) lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of the RMRT-LB method. Thanks to the block-lower triangular relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existing) of the previous MRT-LB models can be obtained, including those based on raw (natural) moment, central moment, Hermite moment, and central Hermite moment. It is also found that when the parameter cs is an adjustable parameter in the standard or rectangular lattice, the present RMRT-LB method becomes a kind of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB models on the DdQq lattice are only its special cases. We also perform some numeric

关键词： Lattice-Boltzmann methods

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ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

arXiv

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arXiv 2023年

作者： Jin, Diancong Sheng, Derui 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 LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this work, we establish the Freidlin-Wentzell large deviations principle (LDP) of the stochastic Cahn-Hilliard equation with small noise, which implies the one-point LDP. Further, we give the one-point LDP of the spatial finite difference method (FDM) for the stochastic Cahn-Hilliard equation. Our main result is the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the Γ-convergence of objective functions, which relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-side Lipschitz, we use the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. This plays an important role in deriving the Γ-convergence of objective functions. Copyright © 2023, The Authors. All rights reserved.

关键词： Finite difference method

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MODERATE DEVIATIONS FOR THE FACILITATED EXCLUSION PROCESS IN EQUILIBRIUM

arXiv

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arXiv 2025年

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

We derive the moderate deviation principles for the fluctuation fields of the facilitated exclusion process (FEP) in one dimension when the process starts from its stationary measure, both in the symmetric and asymmetric cases. The main step is to prove a super-exponential version of the Boltzmann-Gibbs principle, which relies on the logarithmic Sobolev inequality for the FEP. © 2025, CC BY.

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Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

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Physical Review E 2021年第3期104卷 035308-035308页

作者： Xinmeng Chen Zhenhua Chai Jinlong Shang 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

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

关键词： Lattice-Boltzmann methods

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Consistent and conservative phase-field-based lattice Boltzmann method for incompressible two-phase flows

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Physical Review E 2022年第2期106卷 025319-025319页

作者： Chengjie Zhan Zhenhua Chai Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

关键词： Contact line dynamics Multiphase flows Rayleigh-Taylor instability

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