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Optimal rates of convergence and error localization of Gegenbauer projections

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arXiv 2020年

Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and λ ≤ 0 or differentiable and λ ≤ 1, where λ is the parameter in Gegenbauer projections. If the underlying function is analytic and λ > 0 or differentiable and λ > 1, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of nλ and nλ−1, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all λ > −1/2. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral *** Codes 41A10, 41A25 Copyright © 2020, The Authors. All rights reserved.

关键词： Functions

Maxwell-Stefan-theory-based lattice Boltzmann model for diffusion in multicomponent mixtures

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Physical Review E 2019年第2期99卷 023312-023312页

作者： Zhenhua Chai Xiuya Guo Lei 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 Mathematics and Physics China University of Geosciences Wuhan 430074 China

The phenomena of diffusion in multicomponent (more than two components) mixtures are universal in both science and engineering, and from the mathematical point of view, they are usually described by the Maxwell-Stefan (MS)-theory-based diffusion equations where the molar average velocity is assumed to be zero. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures and also perform a Chapman-Enskog analysis to show that the MS continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS-theory-based diffusion equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model and find that the results are in good agreement with the previous work. Besides, the reverse diffusion, osmotic diffusion, and diffusion barrier phenomena are also captured. Finally, compared to the kinetic-theory-based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can still be implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model.

关键词： Lattice-Boltzmann methods

A lattice boltzmann model for the coupled cross-diffusion-fluid system

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arXiv 2021年

作者： Zhan, Chengjie 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 propose a lattice Boltzmann (LB) model for the generalized coupled cross-diffusion-fluid system. Through the direct Taylor expansion method, the proposed LB model can correctly recover the macroscopic equations. The cross diffusion terms in the coupled system are modeled by introducing additional collision operators, which can be used to avoid special treatments for the gradient terms. In addition, the auxiliary source terms are constructed properly such that the numerical diffusion caused by the convection can be eliminated. We adopt the developed LB model to study two important systems, i.e., the coupled chemotaxis-fluid system and the double-diffusive convection system with Soret and Dufour effects. We first test the present LB model through considering a steady-state case of coupled chemotaxis-fluid system, then we analyze the influences of some physical parameters on the formation of sinking plumes. Finally, the double-diffusive natural convection system with Soret and Dufour effects is also studied, and the numerical results agree well with some previous works. © 2021, CC BY.

关键词： Biochemistry

Critical system involving fractional Laplacian

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arXiv 2017年

作者： Zhen, Maoding He, Jinchun Xu, Haoyuan 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 study the following critical system with fractional Laplacian:(Formula Presented) , By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive leas... 详细信息

关键词： Laplace transforms

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

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

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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

New error bounds for Legendre approximations of differentiable functions

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arXiv 2021年

In this paper we present a new perspective on error analysis for Legendre approximations of differentiable functions. We start by introducing a sequence of Legendre-Gauss-Lobatto polynomials and prove their theoretical properties, including an explicit and optimal upper bound. We then apply these properties to derive a new explicit bound for the Legendre coefficients of differentiable functions. Building on this, we establish an explicit and optimal error bound for Legendre approximations in the L2 norm and an explicit and optimal error bound for Legendre approximations in the L∞ norm under the condition that their maximum error is attained in the interior of the interval. Illustrative examples are provided to demonstrate the sharpness of our new results. Copyright © 2021, The Authors. All rights reserved.

关键词： Error analysis

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