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Carleman and observability estimates for stochastic beam equation

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arXiv 2017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 establish a weight identity for stochastic beam equation by means of the multiplier method. Based on this identity, we first establish the global Carleman estimate for the special system with zero initial value and end value, then a revised Carleman estimate for stochastic beam equation is established through a cutoff technique. Finally, we use the revised Carleman estimate to get the required boundary observability estimate. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems

LOCAL SUPERCONVERGENCE OF CONTINUOUS GALERKIN SOLUTIONS FOR DELAY DIFFERENTIAL EQUATIONS OF PANTOGRAPH TYPE

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Journal of Computational Mathematics 2016年第2期34卷 186-199页

作者： Xiuxiu Xu Qiumei Huang Hongtao Chen Beijing Institute for Scientific and Engineering Computing Beijing University of Technology Beijing 100124 China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen 361005 China

This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin solutions under uniform meshes and locate all the superconvergent points based on the supercloseness between the continuous Galerkin solution U and the interpolation Hhu of the exact solution u. The theoretical results are illustrated by numerical examples.

关键词： Pantograph delay differential equations Uniform mesh Continuous Galerkinmethods Supercloseness Superconvergence.

Electrospun nanofibrous membrane for electrodialysis

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Journal of Physics: Conference Series 2019年第1期1209卷

作者： J X Jiang G F Zheng X Wang G Fu J Liu W W Li M F Zhuang SH M Guo Department of Instrumental and Electrical Engineering Xiamen University Xiamen 361102 China School of Mechanical and Manufacturing Engineering Xiamen Institute of Technology Xiamen 361024 China School of Mathematical Sciences Xiamen University Xiamen 361021 China Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing School of Mathematical Sciences Xiamen University Xiamen 361005 China

The direct fabrication of homogenous membrane for electrodialysis has become the research hotspots in the field of water purification. A novel multi-jet electrospinning method with sheath gas was presented to study the fabrication of homogenous electrodialysis membrane. By soaking into sulfuric acid, the reactive exchange groups can be introduced into the electrospun nanofibrous membrane that has been hot pressed. Thanks to the pores among nanofibers, electrospun electrodialysis membrane had advantages of higher porosity, higher water uptake and lower membrane resistance, which promoted the ion transmission through the membrane. The porosity, water uptake and ion exchange capacity (IEC) reduced with the increase of membrane thickness. An electrodialysis separation system was set up to test its separation performance. The results showed that the desalinization ratio of NaCl solution could be reached to 58.15% within 30 minutes. The electrospun nanofibrous membrane for electrodialysis provides a simple and effective way to expand its industrial application.

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A fully discrete energy stable scheme for a phase-field moving contact line model with variabledensities and viscosities

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arXiv 2018年

作者： Zhu, Guangpu Chen, Huangxin Sun, Shuyu Yao, Jun Qingdao266580 China Sch. of Math. Sci. and Fujian Prov. Key Lab. on Math. Modeling and High Perf. Scientific Computing Xiamen University Xiamen361005 China Computational Transport Phenomena Laboratory Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model consists of a Cahn-Hilliard equation, a Navier-Stokes equation and the generalized Navier boundary condition. A scalar auxiliary variable is introduced to transform the nonlinear potential into an equivalent form, allowing the nonlinear potential to be treated effectively and semi-explicitly. A pressure stabilization method is used to decouple the computation of velocity and pressure. Some subtle implicit-explicit treatments are adopted to deal with convention and stress terms. An artificial term is added to balance the explicit nonlinear term originating from the surface energy at fluid-solid interface. We establisharigorous proof of energy stability for the proposed time-marching *** a finite difference methodon staggered grids isusedto spatially discretize the constructed time-marchingscheme. We prove that the fully discrete scheme satisfiesthe discrete energy dissipationlaw. Numerical results demonstrate accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze theevolution of kinetic energy, mixing energy and surface energy at fluid-solid interfaceduring the bubble rising process. Three-dimensional droplet spreading is also investigated onachemically patterned surface. Our numerical simulation accurately predicts the expected energy evolutions and it successfully reproduces expected phenomenathat an oil droplet contractsinwardson a hydrophobiczone and spreadsoutwards quickly on a hydrophiliczone. Copyright © 2018, The Authors. All rights reserved.

关键词： Finite difference method

Jacobi polynomials on the Bernstein ellipse

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arXiv 2017年

作者： Wang, Haiyong Zhang, Lun 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 School of Mathematical Sciences Shanghai Key Laboratory for Contemporary Applied Mathe-matics Fudan University Shanghai200433

In this paper, we are concerned with Jacobi polynomials P(α,β)n(x) on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of P(α,β)n(x) is derived in the variable of parametrization. This formula further allows us to show that the maximum value of |P(α,β)n(z)| over the Bernstein ellipse is attained at one of the endpoints of the major axis if α+β ≥ -1. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., α = β), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.65N35, 65D05, 41A05, 41A25 Copyright © 2017, The Authors. All rights reserved.

关键词： Polynomials

Lattice boltzmann modeling of wall-bounded ternary fluid flows

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arXiv 2017年

作者： Liang, Hong Xu, Jiangrong Chen, Jiangxing Chai, Zhenhua Shi, Baochang Department of Physics Hangzhou Dianzi University Hangzhou310018 China 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 wetting boundary scheme used to describe the interactions among ternary fluids and solid is proposed in the framework of the lattice Boltzmann method. This scheme for three-phase fluids can preserve the reduction consistency property with the diphasic situation such that it could give physically relevant results. Combining this wetting boundary scheme and the lattice Boltzmann (LB) ternary fluid model based on the multicomponent phase-field theory, we simulated several ternary fluid flow problems involving solid substrate, including the spreading of binary drops on the substrate, the spreading of a compound drop on the substrate, and the shear of a compound liquid drop on the substrate. The numerical results are found to be good agreement with the analytical solutions or some available results. Finally, as an application, we use the LB model coupled with the present wetting boundary scheme to numerically investigate the impact of a compound drop on a solid circular cylinder. It is found that the dynamics of a compound drop can be remarkably influenced by the wettability of the solid surface and the dimensionless Weber number. Copyright © 2017, The Authors. All rights reserved.

关键词： Circular cylinders

Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows

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arXiv 2017年

作者： Liang, Hong Xu, Jiangrong Chen, Jiangxing Wang, Huili Chai, Zhenhua Shi, Baochang Department of Physics Hangzhou Dianzi University Hangzhou310018 China 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 present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A novel forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and Spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. At last, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature. Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

A note on optimal decay rates for the axisymmetric D-solutions to the steady Navier-Stokes equations

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arXiv 2017年

作者： Weng, Shangkun Xie, Chunjing School of Mathematics and Statistics Computational Science Hubei Key Laboratory Wuhan University Wuhan Hubei Province430072 China Department of Mathematics Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai200240 China

In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in R3. We obtain the optimal decay rate |u(x)| | xC|+1 for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component ua, that is, either O(+11) +1or |ue(r,z)| (C1r)3, where p = Vr2+ z2. In the latter case, we can further deduce that the other two components of the velocity field also attain the optimal decay rates: |ur(r, z)| + |uz(r, z)| C+1-. We do not require any small assumptions on the forcing *** Codes 76D05 Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

Correction to: A Linearized Local Conservative Mixed Finite Element Method for Poisson–Nernst–Planck Equations

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Journal of scientific computing 2018年第2期77卷 818-818页

作者： Gao, Huadong Sun, Pengtao School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan People’s Republic of China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China Department of Mathematical Sciences University of Nevada Las Vegas Las Vegas USA

The original version of this article contained a mistake. There are error in line breaks in Eqs. 4.3 and 4.4 and the word “quad” was included inadvertently in Eq. 4.4.

The original version of this article contained a mistake. There are error in line breaks in Eqs. 4.3 and 4.4 and the word “quad” was included inadvertently in Eq. 4.4.

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