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Numerical Method for Non-Linear conservation laws: Inviscid Burgers Equation

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Journal of Applied Mathematics and Physics 2021年第6期9卷 1351-1363页

作者： R. Hemel M. T. Azam M. S. Alam Department of Mathematics Jahangirnagar University Dhaka Bangladesh

This paper deals with the Burgers equation which is the most common model used in the nonlinear conservation laws. Here the theoretical aspect of conservation law is discussed by using inviscid Burgers equation. At first, we introduce the general non-linear conservation law as a partial differential equation and its solution procedure by the method of characteristic. Next, we present the weak solution of the problem with entropy condition. Taking into account shock wave and rarefaction wave, the Riemann problem has also been discussed. Finally, the finite volume method is considered to approximate the numerical solution of the inviscid Burgers equation with continuous and discontinuous initial data. An illustration of the problem is provided by some examples. Moreover, the Godunov method provides a good approximation for the problem.

关键词： conservation law Inviscid Burgers Equation Shock Wave Rarefaction Wave Finite Volume Method

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrdinger system

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Chinese Physics B 2013年第6期22卷 135-140页

作者：蔡加祥王雨顺 Jiangsu Key Laboratory for NSLSCS School of Mathematics ScienceNanjing Normal University School of Mathematics Science Huaiyin Normal University

We derive a new method for a coupled nonlinear Schr/Sdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.

关键词： Schroedinger equation Fourier pseudospectral method conservation law energy

Noether symmetries of discrete mechanico-electrical systems

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Chinese Physics B 2008年第12期17卷 4354-4360页

作者：傅景礼陈本永谢凤萍 Institute of Mathematical Physics Zhejiang Sci-Tech University Institute of Mechanical and Automatism Control Engineering Zhejiang Sci-Tech University

This paper focuses on studying Noether symmetries and conservation laws of the discrete mechanico-electricM systems with the nonconservative and the dissipative forces. Based on the invariance of discrete Hamilton action of the systems under the infinitesimal transformation with respect to the generalized coordinates, the generalized electrical quantities and time, it presents the discrete analogue of variational principle, the discrete analogue of Lagrange-Maxwell equations, the discrete analogue of Noether theorems for Lagrange Maxwell and Lagrange mechanico-electrical systems. Also, the discrete Noether operator identity and the discrete Noether-type conservation laws are obtained for these systems. An actual example is given to illustrate these results.

关键词： Noether symmetry conservation law discrete mechanico-electrical system dissipative function

Riccati-type Backlund transformations of nonisospectral and generalized variable-coefficient KdV equations

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Chinese Physics B 2014年第3期23卷 186-192页

作者：杨云青王云虎李昕程雪苹 School of Mathematics Physics and Information Science Zhejiang Ocean University Shanghai Key Laboratory of Trustworthy Computing East China Normal University School of Mathematics and Statistics Changshu Institute of Technology

We extend the method of constructing Bgcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg-de Vries （KdV） equations as examples, their Backlund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Es- pecially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.

关键词： Baicklund transformation Lax pair conservation law Cole-Hopf transformation

Binary Bell polynomial application in generalized(2+1)-dimensional KdV equation with variable coefficients

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Chinese Physics B 2011年第11期20卷 34-40页

作者：张翼魏薇薇程腾飞宋洋 Department of Mathematics Zhejiang Normal University

In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized （2＋1）-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.

关键词： binary Bell polynomial bilinear Backlund transformation Lax pair conservation law

Multi-symplectic method for generalized fifth-order KdV equation

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Chinese Physics B 2008年第11期17卷 3923-3929页

作者：胡伟鹏邓子辰 School of Mechanics Civil Engineering and Architecture Northwestern Polytechnical University State Key Laboratory of Structural Analysis of Industrial Equipment Dalian University of Technology

This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.

关键词： generalized fifth-order KdV equation multi-symplectic travelling wave solution conservation law

A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL FOR BI-DIRECTIONAL PEDESTRIAN FLOWS

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Acta Mathematica Scientia 2009年第6期29卷 1541-1555页

作者： Yanqun Jiang Tao Xiong S.C. Wong Chi-Wang Shu Mengping Zhang Peng Zhang William H.K. Lam Department of Mathematics University of Science and Technology of China Hefei 230026 China Department of Civil Engineering The University of Hong Kong Hong Kong China Division of Applied Mathematics Brown University Providence RI 02912 U.S.A. Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China Department of Civil and Structural Engineering The Hong Kong Polytechnic University Hong Kong China

In this paper, a reactive dynamic user equilibrium model is extended to simulate two groups of pedestrians traveling on crossing paths in a continuous walking facility. Each group makes path choices to minimize the travel cost to its destination in a reactive manner based on instantaneous information. The model consists of a conservation law equation coupled with an Eikonal-type equation for each group. The velocity-density relationship of pedestrian movement is obtained via an experimental method. The model is solved using a finite volume method for the conservation law equation and a fast-marching method for the Eikonal-type equation on unstructured grids. The numerical results verify the rationality of the model and the validity of the numerical method. Based on this continuum model, a number of results, e.g., the formation of strips or moving clusters composed of pedestrians walking to the same destination, are also observed.

关键词： pedestrian flows conservation law Eikonal-type equation density-velocity relationship finite volume method fast marching method unstructured grids

A conservative Fourier pseudospectral algorithm for the nonlinear Schrodinger equation

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Chinese Physics B 2014年第12期23卷 21-29页

作者：吕忠全张鲁明王雨顺 College of Science Nanjing University of Aeronautics and Astronautics College of Science Nanjing Forestry University School of Mathematical Sciences Nanjing Normal University

In this paper, we derive a new method for a nonlinear Schrodinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ｜｜·｜｜2 norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.

关键词： Fourier pseudospectral method Schrdinger equation conservation law convergence

Soliton dynamics in planar ferromagnets and anti-ferromagnets

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Journal of Zhejiang University Science 2003年第5期4卷 503-510页

作者： LINFang-hua SHATAHJalal CourantInstitute NewYorkUniversityNY10012USA

The aim of this paper is to present a rigorous mathematical proof of the dynamical laws for the topological solitons( magnetic vortices) in ferromagnets and anti-ferromagnets. It is achieved through the conservation l... 详细信息

关键词： Magnetic vortices Topological vorticity conservation law Soliton dynamics