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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

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

SUPERCONVERGENCE ANALYSIS FOR TIME-FRACTIONAL DIFFUSION equations WITH NONCONFORMING MIXED FINITE ELEMENT METHOD

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Journal of Computational Mathematics 2019年第4期37卷 488-505页

作者： Houchao Zhang Dongyang Shi School of Mathematics and Statistics Pingdingshan University Pingdingshan 467000 China School of Mathematics and Statistics Zhengzhou University Zhengzhou 450001 China

In this paper, a fully discrete scheme based on the LI approximation in temporal direction for the fractional derivative of order in (0,1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order O(h^2)of EQ1^rot element (see Lemma 2.3). Then, by using the proved character of EQ1^rot element, we present the superconvergent estimates for the original variable u in the broken H^1-norm and the flux →p =△u in the (L^2)^2-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

关键词： Nonconforming MFEM LI method Time-fractional diffusion equations Superconvergence

Introduction to the Special Issue on Numerical Methods for Differential and Integral equations

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Computer Modeling in Engineering & Sciences 2020年第6期123卷 913-915页

作者： Şuayip Yüzbaşı Kamel Al-Khaled Nurcan BaykuşSavaşaneril Devendra Kumar Department of Mathematics Faculty of ScienceAkdeniz UniversityAntalyaTr-07058Turkey Department of Mathematics and Statistics Jordan University of Science and TechnologyIrbid22110Jordan Izmir Vocational School Dokuz Eylül UniversityİzmirTurkey Department of Mathematics University of RajasthanJaipur302004India

Differential and integral equations have a major role in many applied areas of science and engineering,For example,physics,chemistry,astronomy,biology,mechanics,electronic,economics,potential theory,electrostatics.So,the obtaining of solutions of these equations is important.Since the mentioned equations are usually difficult to solve analytically,numerical methods are required.Therefore,this Special Issue contributes to new numerical methods for solving the above-mentioned equations and thus,many problems in science and engineering can be solved using new numerical methods analyzed in this special issue.

关键词： mentioned equations Issue

C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order equations

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Communications on Applied Mathematics and Computation 2019年第3期1卷 403-434页

作者： Huiyuan Li Weikun Shan Zhimin Zhang State Key Laboratory of Computer Science/Laboratory of Parallel Computing Institute of SoftwareChinese Academy of SciencesBeijing 100190China School of Science Henan University of TechnologyZhengzhou 450001HenanChina Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

关键词： Quadrilateral spectral element method Fourth-order equations Mapped polynomials C1-conforming basis Polynomial inclusion Completeness

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

EFFICIENT AND ACCURATE NUMERICAL METHODS FOR LONG-WAVE SHORT-WAVE INTERACTION equations IN THE SEMICLASSICAL LIMIT REGIME

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Journal of Computational Mathematics 2019年第5期37卷 645-665页

作者： Tingchun Wang Xiaofei Zhao Mao Peng Peng Wang School of Mathematics and Statistics Nanjing University of Information Science and TechnologyNanjing 210044China Department of mathematics National University of SingaporeSingapore 119076 Institute of Mathematics Jilin UniversityChangchun 130012China

This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on:(i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations;(ii) the ap-plication of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives;(iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in L^1. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstrate the behavior of nonlinear dispersive waves in the semi-classical limit regime.

关键词： Long-wave short-wave interaction equations Semiclassical limit Time-splitting method Spectral method Compact finite difference method Conservative properties

Numerical study of edge waves using extended Boussinesq equations

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Water Science and Engineering 2017年第4期10卷 295-302页

作者： Gang Wang Zhong-bin Sun Jun-liang Gao Xiao-zhou Ma Jiangsu Key Laboratory of Coast Ocean Resources Development and Environment Security(Hohai University) Nanjing 210098China Nanjing Hydraulic Research Institute Nanjing 210029China School of Naval Architecture and Ocean Engineering Jiangsu University of Science and TechnologyZhenjiang 212003China State Key Laboratory of Coastal and Offshore Engineering Dalian University of TechnologyDalian 116023China

An edge wave numerical model was developed based on extended Boussinesq equations with the internal wave-generation method. The form of edge waves near a seawall was chosen as the input signal in order to avoid treatment of the moving shoreline on a sloping beach. As there was an energy transfer between different edge wave modes, not only the target mode but also other modes appeared in the simulations. Due to the nonlinear effect, the simulation results for mode-0 edge waves were slightly modulated by mode-1 and mode-2 waves. As the magnitudes of these higher-mode waves are not significantly related to those of the target mode, the internal wave-generation method in Boussinesq equations can produce high-quality edge waves. The numerical model was used to investigate the nonlinear properties of standing edge waves, and the numerical results were in strong agreement with theory.

关键词： Edge waves Boussinesq equations Internal wave-generation method Nonlinear wave interaction