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Discrete effects on some boundary schemes of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations

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arXiv 2019年

作者： Wu, Yao Zhao, Yong Chai, Zhenhua Shi, Baochang 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, we perform a more general analysis on the discrete effects of some boundary schemes of the popular one- to three-dimensional DnQq multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation (CDE). Investigated boundary schemes include anti-bounce-back(ABB) boundary scheme, bounce-back(BB) boundary scheme and non-equilibrium extrapolation(NEE) boundary scheme. In the analysis, we adopt a transform matrix M constructed by natural moments in the evolution equation, and the result of ABB boundary scheme is consistent with the existing work of orthogonal matrix M. We also find that the discrete effect does not rely on the choice of transform matrix, and obtain a relation to determine some of the relaxation-time parameters which can be used to eliminate the numerical slip completely under some assumptions. In this relation, the weight coefficient is considered as an adjustable parameter which makes the parameter adjustment more flexible. The relaxation factors associated with second moments can be used to eliminate the numerical slip of ABB boundary scheme and BB boundary scheme while the numerical slip can not be eliminated of NEE boundary scheme. Furthermore, we extend the relations to complex-valued CDE, several numerical examples are used to test the relations. Copyright © 2019, The Authors. All rights reserved.

关键词： Diffusion in liquids

Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system based incompressible Navier-Stokes equations

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arXiv 2021年

作者： Chai, Zhenhua Shi, Baochang Zhan, Chengjie 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, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations (NSEs) which are considered as the coupled convection-diffusion equations (CDEs). Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of distribution function. Then in the framework of present MDF-LBM, we develop a locally computational scheme for the velocity gradient where the first-order moment of the non-equilibrium distribution is used, this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress and vorticity. Finally, we also conduct some simulations to test the MDF-LBM, and find that the numerical results not only agree with some available analytical and numerical solutions, but also have a second-order convergence rate in space. © 2021, CC BY.

关键词： Kinetic theory

Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

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arXiv 2024年

作者： Wang, Haiyong Wu, Menghan 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

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to [0, 1/2]. In this paper we consider the construction of generalized Gauss-Radau quadrature rules for Hankel transform. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss-Radau quadrature rules for Hankel transform of integer order can be constructed with theoretical guarantees. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to confirm our *** Codes 65R10, 65D32 © 2024, CC BY-NC-ND.

关键词： Integral equations

Macroscopic finite-difference scheme based on the mesoscopic regularized lattice-Boltzmann method

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Physical Review E 2024年第2期109卷 025301-025301页

作者： Xi Liu Ying Chen 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 paper, we develop a macroscopic finite-difference scheme from the mesoscopic regularized lattice Boltzmann (RLB) method to solve the Navier-Stokes equations (NSEs) and convection-diffusion equation (CDE). Unlike the commonly used RLB method based on the evolution of a set of distribution functions, this macroscopic finite-difference scheme is constructed based on the hydrodynamic variables of NSEs (density, momentum, and strain rate tensor) or macroscopic variables of CDE (concentration and flux), and thus shares low memory requirement and high computational efficiency. Based on an accuracy analysis, it is shown that, the same as the mesoscopic RLB method, the macroscopic finite-difference scheme also has a second-order accuracy in space. In addition, we would like to point out that compared with the RLB method and its equivalent macroscopic numerical scheme, the present macroscopic finite-difference scheme is much simpler and more efficient since it is only a two-level system with macroscopic variables. Finally, we perform some simulations of several benchmark problems, and find that the numerical results are not only in agreement with analytical solutions, but also consistent with the theoretical analysis.

关键词： Complex fluids Lattice-Boltzmann methods Navier-Stokes equation

Analysis of error localization of Chebyshev spectral approximations

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arXiv 2021年

作者： 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

Chebyshev spectral methods are widely used in numerical computations. When the underlying function has a singularity, it has been observed by L. N. Trefethen in 2011 that its Chebyshev interpolants exhibit an error localization property, that is, their errors in a neighborhood of the singularity are obviously larger than elsewhere. In this paper, we first present a pointwise error analysis for Chebyshev projections of functions with a singularity and prove that the rate of convergence of Chebyshev projections of degree n at each point away from the singularity is one power of n faster than that of at the singularity. This gives a rigorous justification for the error localization of Chebyshev projections. We then extend the framework of our analysis to Chebyshev interpolants, Chebyshev spectral differentiations and Legendre projections and justify their error localization using similar arguments. As a result, we find that Chebyshev spectral differentiations converge faster than their best counterparts except in a neighborhood of the singularity and, in the particular case where the singularity is located in the interior of interval, they converge even faster than their best counterparts in the maximum *** Codes 41A10, 41A25, 41A50 © 2021, CC BY-NC-SA.

关键词： Numerical methods

CENTRAL LIMIT THEOREM FOR TEMPORAL AVERAGE OF BACKWARD EULER–MARUYAMA METHOD

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arXiv 2023年

作者： Diancong, J.I.N. 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

This work focuses on the temporal average of the backward Euler–Maruyama (BEM) method, which is used to approximate the ergodic limit of stochastic ordinary differential equations with super-linearly growing drift coefficients. We give the central limit theorem (CLT) of the temporal average, which characterizes the asymptotics in distribution of the temporal average. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM method. For the case that the deviation order equals to the optimal strong order, the CLT is established via the Poisson equation associated with the generator of original equations. Numerical experiments are performed to illustrate the theoretical results. Copyright © 2023, The Authors. All rights reserved.

关键词： Poisson equation

Generalized lattice boltzmann method: modeling, analysis, and elements

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arXiv 2019年

作者： Shi, Baochang Chai, Zhenhua 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, we first present a unified framework for the modelling of generalized lattice Boltzmann method (GLBM). We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes equations and nonlinear convection-diffusion equations from the GLBM, and show that from mathematical point of view, these four analysis methods are equivalent to each other. Finally, we give some elements that are needed in the implementation of the GLBM, and also find that some available LB models can be obtained from this GLBM. Copyright © 2019, The Authors. All rights reserved.

关键词： Nonlinear equations

Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

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arXiv 2018年

作者： Zhen, Maoding He, Jinchun Xu, Haoyuan Yang, Meihua 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, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent (formula presented), where (−∆)s is the fractional Laplacian, 0 2s, λ ∗−1 and 2∗ = N2−N2s is the Sobolev critical exponent. By using the Nehari manifold, we show that there exists a µ0 ∈ (0, 1), such that when 0 µ0, there exists a λµ,ν (formula presented) such that if λ > λµ,ν, the system has a positive ground state solution, if λ µ,ν, the system has no ground state solution. Copyright © 2018, The Authors. All rights reserved.

关键词： Laplace transforms

Convergence analysis of Laguerre approximations for analytic functions

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arXiv 2023年

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n converge at the root-exponential rate O(exp(−2ρ√n)) with ρ > 0 when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at z = −ρ2. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical *** Codes 41A25, 41A10 © 2023, CC BY-NC-SA.

关键词： Laplace transforms