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An equivalent form of the Navier-Stokes Equations

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Acta Mechanica Sinica 2023年第2期39卷 165-166页

作者： Bo-Hua Sun School of Civil Engineering&Institute of Mechanics and Technology Xi'an University of Architecture and TechnologyXi'an 710055China

The Navier-Stokes Equations of incompressible flow can be expressed as follows[1-8]:■v/■t+■·■u=-1/ρ■ρ+v■^(2)v,(1)■·v=0(2)Equation(1)is the momentums equation and Eq.(2)is the mass conservation equation.

关键词： equation. Navier Equations

The Multi-scale Method for Solving Nonlinear Time Space Fractional Partial Differential Equations

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IEEE/CAA Journal of Automatica Sinica 2019年第1期6卷 299-306页

作者： Hossein Aminikhah Mahdieh Tahmasebi Mahmoud Mohammadi Roozbahani the Department of Applied Mathematics University of Guilan the Department of Applied Mathematics Faculty of Mathematical SciencesTarbiat Modares University

In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential Equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of Equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.

关键词： Adams fractional method B-spline wavelets multi-scale method nonlinear fractional partial differential Equations

Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer Equations

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Applied Mathematics and Mechanics(English Edition) 2019年第6期40卷 889-910页

作者： Demin LIU Kaitai LI College of Mathematics and System Sciences Xinjiang UniversityUrumqi 830046China School of Mathematics and Statistics Xi’an Jiaotong UniversityXi’an 710049China

In this work, the two-dimensional convective Brinkman-Forchheimer equa- tions are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.

关键词： Brinkman-Forchheimer Equations finite element method Newton iteration

A Study of Hyperbolicity of Kinetic Stochastic Galerkin System for the Isentropic Euler Equations with Uncertainty

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Chinese Annals of Mathematics,Series B 2019年第5期40卷 765-780页

作者： Shi JIN Ruiwen SHU School of Mathematical Sciences Institute of Natural SciencesMOE-LSEC and SHL-MACShanghai Jiao Tong University800 Dongchuan Rd.Shanghai 200240China Department of Mathematics University of MarylandCollege Park4176 Campus Dr.College ParkMD 20742USA.

The authors study the fluid dynamic behavior of the stochastic Galerkin(SG for short) approximation to the kinetic Fokker-Planck equation with random uncertainty.While the SG system at the kinetic level is hyperbolic, its fluid dynamic limit, as the Knudsen number goes to zero and the underlying kinetic equation approaches to the uncertain isentropic Euler Equations, is not necessarily hyperbolic, as will be shown in the case study fashion for various orders of the SG approximations.

关键词： Hyperbolic Equations Uncertainty quantification Stochastic Galerkin methods

Pseudo Asymptotically Periodic Solutions for Volterra Difference Equations of Convolution Type

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Chinese Annals of Mathematics,Series B 2019年第4期40卷 501-514页

作者： Zhinan XIA Department of Applied Mathematics Zhejiang University of Technology

In this paper, the author studies the existence and uniqueness of discrete pseudo asymptotically periodic solutions for nonlinear Volterra difference Equations of convolution type, where the nonlinear perturbation is considered as Lipschitz condition or non-Lipschitz case, respectively. The results are a consequence of application of different fixed point theorems, namely, the contraction mapping principle, the Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

关键词： Pseudo asymptotically periodic function Volterra difference Equations Contraction mapping principle Leray-Schauder alternative theorem

Local Well-posedness for Linearized Degenerate MHD Boundary Layer Equations in Analytic Setting

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Acta Mathematica Sinica,English Series 2019年第8期35卷 1402-1418页

作者： Ya Jun LI Wen Dong WANG School of Mathematical Sciences Dalian University of Technology

In this note we derive MHD boundary layer Equations according to viscosity and resistivity coefficients. Especially, when these viscosity and resistivity coefficients are of different orders, it leads to degenerate MH... 详细信息

关键词： MHD Equations degenerate boundary layer Equations local well-posedness analytic spaces

Stability of Rarefaction Wave to the 1-D Piston Problem for the Pressure-Gradient Equations

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Chinese Annals of Mathematics,Series B 2019年第2期40卷 161-186页

作者： Min DING Department of Mathematics School of Science Wuhan University of Technology

The 1-D piston problem for the pressure gradient Equations arising from the flux-splitting of the compressible Euler Equations is considered. When the total variations of the initial data and the velocity of the piston are both sufficiently small, the author establishes the global existence of entropy solutions including a strong rarefaction wave without restriction on the strength by employing a modified wave front tracking method.

关键词： Piston problem Pressure gradient Equations Rarefaction wave Wave front tracking method Interaction of waves

A GENERAL CLASS OF ONE-STEP APPROXIMATION FOR INDEX-1 STOCHASTIC DELAY-DIFFERENTIAL-ALGEBRAIC *Equations

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Journal of Computational Mathematics 2019年第2期37卷 151-169页

作者： Tingting Qin Chengjian Zhang 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

This paper develops a class of general one-step discretization methods for solving the index-1 stochastic delay differential-algebraic Equations. The existence and uniqueness theorem of strong solutions of index-1 Equations is given. A strong convergence criterion of the methods is derived, which is applicable to a series of one-step stochastic numerical methods. Some specific numerical methods, such as the Euler-Maruyama method, stochastic ^-methods, split-step ^-methods are proposed, and their strong convergence results are given. Numerical experiments further illustrate the theoretical results.

关键词： Stochastic delay differential-algebraic Equations One-step discretization schemes Strong convergence

A DECOUPLING TWO-GRID METHOD FOR THE STEADY-STATE POISSON-NERNST-PLANCK Equations

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Journal of Computational Mathematics 2019年第4期37卷 556-578页

作者： Ying Yang Benzhuo Lu Yan Xie School of Mathematics and Computing Science Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation Guilin University of Electronic Technology Guilin 541004 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing the National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Poisson-Nernst-Planck Equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. Two kinds of two-grid finite element algorithms are proposed to decouple the steady-state Poisson-Nernst-Planck Equations by coarse grid finite element approximations. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid algorithms for solving Poisson-Nernst-Planck Equations.

关键词： Poisson-Nernst-Planck Equations Two-grid finite element method Decoupling method Error analysis Gummel iteration