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

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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

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

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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

THE DECAY ESTIMATES FOR MAGNETOHYDRODYNAMIC EQUATIONS WITH COULOMB FORCE

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Acta Mathematica Scientia 2020年第6期40卷 1928-1940页

作者： Wenxuan ZHENG Zhong TAN School of Mathematical Sciences Xiamen UniversityXiamen 361005China School of Mechanical and Electronic Engineering Tarim UniversityAlar 843300China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing Xiamen UniversityXiamen 361005China

In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb *** spectral analysis and energy methods,we obtain the optimal time decay estimate of the *** show that the global classi... 详细信息

关键词： lower convergence rates upper decay rates spectral analysis energy method

Linearly Implicit and High-Order Energy-Preserving Schemes for Highly Oscillatory Hamiltonian Systems

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SSRN

SSRN 2022年

作者： Li, Dongfang Li, Xiaoxi Zhang, Zhimin 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 Mathematics Wayne State University DetroitMI48202 United States

In this paper, a family of novel energy-preserving schemes are presented for numerically solving highly oscillatory Hamiltonian systems. These schemes are constructed by using the relaxation idea in the extrapolated Runge–Kutta (ERK) methods. After obtaining the relaxation parameter, it is shown that the methods can be arbitrarily high order accurate, linearly implicit and keep the original discrete energy conserved, while the previous energy-preserving schemes for highly oscillatory Hamiltonian systems are usually fully implicit. Numerical comparisons with various typical energy-preserving schemes are presented. The numerical results show that the proposed schemes are highly competitive and effective. © 2022, The Authors. All rights reserved.

关键词： Runge Kutta methods

Splitting Method for Stochastic Navier-Stokes Equations

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arXiv 2025年

作者： Zhu, Jie Zhu, Yujun Ming, Ju Gunzburger, Max D. School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Department of Scientific Computing Florida State University TallahasseeFL32304 United States

This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS component and a stochastic equation. We rigorously analyze the proposed splitting method from the perspectives of equivalence, stability, existence and uniqueness of the solution. We also propose a modified splitting scheme, which simplified the stochastic equation by omitting its nonlinear terms. A detailed analysis of the solution properties for this modified approach is provided. Additionally, we discuss the statistical errors with both the original splitting format and the modified scheme. Our theoretical and numerical studies demonstrate that the equivalent splitting scheme exhibits significantly enhanced stability compared to the original stochastic NS equations, enabling more effective handling of nonlinear characteristics. Several numerical experiments were performed to compare the statistical errors of the splitting method and the modified splitting method. Notably, the deterministic NS equation in the splitting method does not require repeated solving, and the stochastic equation in the modified scheme is free of nonlinear terms. These features make the modified splitting method particularly advantageous for large-scale computations, as it significantly improves computational efficiency without compromising accuracy. Copyright © 2025, The Authors. All rights reserved.

关键词： Navier Stokes equations

A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

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Communications on Applied Mathematics and Computation 2020年第2期2卷 179-199页

作者： Junying Cao Ziqiang Wang Chuanju Xu School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

关键词： Caputo-Fabrizio derivative Fractional diferential equations High-order numerical scheme

The optimal displacement of immiscible two-phase fluids in a pore doublet

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arXiv 2022年

作者： Shan, Fang Chai, Zhenhua Shi, Baochang Zhao, Meng 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 displacement of multiphase fluid flow in a pore doublet is a fundamental problem, and is also of importance in understanding of the transport mechanisms of multiphase flows in the porous media. During the displacement of immiscible two-phase fluids in the pore doublet, the transport process is not only influenced by the capillary and viscous forces, but also affected by the channel geometry. In this paper, we first present a mathematical model to describe the two-phase fluid displacement in the pore doublet where the effects of capillary force, viscous force and the geometric structure are included. Then we derive an analytical solution of the model for the first time, and find that the displacement process is dominated by the capillary number, the viscosity ratio and the radius ratio. Furthermore, we define the optimal displacement that the wetting fluids in two daughter channels break through the branches simultaneously (both of them have the same breakthrough time), and also obtain the critical capillary number corresponding to the optimal displacement, which is related to the radius ratio of two daughter channels and viscosity ratio of two immiscible fluids. Finally, it is worthy noting that the present analytical results on the displacement in the pore doublet can be used to explain and understand the phenomenon of preferential imbibition or preferential flow in porous media. © 2022, CC BY.

关键词： Porous materials

Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations

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Physical Review E 2021年第1期104卷 015312-015312页

作者： Yuxin Lin Ning Hong Baochang Shi Zhenhua Chai School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China School of General Education Wuchang University of Technology Wuhan 430223 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω0 and the relaxation parameters s1 and s2 corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations and find that the numerical results are consistent with our theoretical analysis.

关键词： Lattice-Boltzmann methods

Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions

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arXiv 2023年

作者： Lin, Bin Mao, Zhiping Wang, Zhicheng Karniadakis, George Em School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen361005 China Laboratory of Ocean Energy Utilization Ministry of Education School of Energy and Power Engineering Dalian University of Technology Dalian China Division of Applied Mathematics Brown University ProvidenceRI02906 United States

Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has been introduced and extensively employed in approximating solution of PDEs. Nevertheless, to solve problems consisting of sharp solutions poses a significant challenge when employing these two approaches. To address this issue, we propose in this work a novel framework termed Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize DeepONet to learn the solution operator for a set of smooth problems relevant to the PDEs characterized by sharp solutions. Subsequently, we integrate the pre-trained DeepONet with PINN to resolve the target sharp solution problem. We showcase the efficacy of OL-PINN by successfully addressing various problems, such as the nonlinear diffusion-reaction equation, the Burgers equation and the incompressible Navier-Stokes equation at high Reynolds number. Compared with the vanilla PINN, the proposed method requires only a small number of residual points to achieve a strong generalization capability. Moreover, it substantially enhances accuracy, while also ensuring a robust training process. Furthermore, OL-PINN inherits the advantage of PINN for solving inverse problems. To this end, we apply the OL-PINN approach for solving problems with only partial boundary conditions, which usually cannot be solved by the classical numerical methods, showing its capacity in solving ill-posed problems and consequently more complex inverse problems. Copyright © 2023, The Authors. All rights reserved.

关键词： Partial differential equations