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FINITE DIFFERENCE METHOD WITH NONUNIFORM MESHES FOR QUASILINEAR PARABOLIC SYSTEMS

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Journal of computational mathematics 1997年第4期15卷 289-310页

作者： Yu-lin Zhou (Labomtory of computational physics, Center of Nonlinear Studies, {2., beijing, china) 北京应用物理和计算数学所北京

The analysis of the finite difference schemes with nonuniform meshes for the problems of partial differential equations is extremely rare even for very simple problems and even for the method of fully heuristic character. In the present work the boundary value problem for quasilinear parabolic system is solved by the finite difference method with nonuniform meshes. By using of the interpolation formulas for the spaces of discrete functions with unequal meshsteps and the method of a priori estimation for the discrete solutions of finite difference schemes with nonuniform meshes, the absolute and relative convergence of the discrete solutions of the finite defference scheme are proved. The limiting vector function is just the unique generalized solution of the original problem for the parabolic system.

关键词： Boundary value problems Convergence of numerical methods Estimation Heuristic methods Interpolation Partial differential equations

A numerical study on Rayleigh-Taylor instability of aluminum plates driven by detonation

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Science china(physics,Mechanics & Astronomy) 2010年第2期53卷 195-198页

作者： HE ChangJiang,ZHOU HaiBing & HANG YiHong institute of applied physics and computational mathematics,laboratory of {2. physics,beijing 1.0088.china institute of applied physics and computational mathematics Laboratory of Computational Physics Beijing China

The authors,using elastic-plastic hydrodynamic code,present the Rayleigh-Taylor (RT) instability of Al plates driven by high-explosive detonation. Our numerical study assumes the material is fluid,or it is an elastic-plastic solid,and we compare the results of these simulations with the experimental data. For the numerical simulation of Rayleigh-Taylor instability of the metal driven by high-explosive detonation,the elastic-plastic effect must be assumed. The result of the simulation is different from the experiment,using only equation of state. However,the growth of perturbation agrees well with the measured growth under the second assumption. There is a cutoff wavelength for RT instability of the metal. The growth of perturbation is stable for short wavelength. The growth increases rapidly as the wavelength increases.

关键词： interface instability elastic-plastic numerical simulation

Ab initio study of a Y-doped Σ31 grain boundary in alumina

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Science china(physics,Mechanics & Astronomy) 2008年第11期51卷 1607-1615页

作者： CHEN Jun, XU Yun, CHEN DongQuan & ZHANG JingLin laboratory of computational {2., institute of applied physics and computational mathematics, beijing 1.0088. china laboratory of computational {2. Institute of Applied Physics and Computational Mathematics Beijing China

The atomic structures and energetics of clean and Y-doped general grain boundary (GB) Σ31/(0001) models in α-Al2.3 are studied by a series of high precision ab initio calculations. A large supercell with 700 atoms and periodic boundary conditions is adopted for undoped and Y-doped GB with different substitution sites and con-centrations. It is shown that Y atoms preferably segregate to the central column of the 7-member Al ring. This is explained as more favorable bond formation for Y in this position and lower GB energy. The calculated GB formation energy for the clean and Y-doped cases is respectively 3.99 and 3.67 J/m2. On the average, the GB region in Σ31 has a slightly lower charge density than the bulk crystalline region. In addtition, the GB induces a long ranged asymmetric electrostatic potential distri-bution on each side of the grain boundary.

关键词： alumina grain boundary Y-doped ab initio

LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF DIFFERENCE SCHEME FOR A SEMILINEAR PARABOLIC EQUATION (Ⅱ)

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Journal of computational mathematics 1998年第5期16卷 571-576页

作者： Hui Feng(Shenzhen University Normal College, Shenzhen 518060, china Laboratorp of computational {2., institute of applied physics and computational mathematics,P.O.Box 8009, beijing 1.0088. china )Long-jnn Shen(laboratory of computational {2., In 深圳大学师范学院广东深圳

In this paper we prove the solution of explicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the relevant nonlinear stationary problem as t→∞. For nonlinear parabolic problem, we obtain the long time asymptotic behavior of its discrete solution which is analogous to that of its continuous solution. For simplicity, we discuss one-dimensional problem.

关键词： Asymptotic behavior Explicit difference scheme Semilinear parabolic equation.

The unconditional stable and convergent difference methods with intrinsic parallelism for quasilinear parabolic systems

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Science china mathematics 2004年第3期47卷 453-472页

作者： ZHOU Yulin YUAN Guangwei SHEN Longjun laboratory of computational {2., Center of Nonlinear Studies, institute of applied physics and computational mathematics, beijing 1.0088. china laboratory of computational {2. Center of Nonlinear Studies Institute of Applied Physics and Computational Mathematics Beijing China

A kind of the general finite difference schemes with intrinsic parallelism for the boundary value problem of the quasilinear parabolic system is studied without assuming heuristically that the original boundary value problem has the unique smooth vector solution. By the method of a priori estimation of the discrete solutions of the nonlinear difference systems, and the interpolation formulas of the various norms of the discrete functions and the fixed-point technique in finite dimensional Euclidean space, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general finite difference schemes with intrinsic parallelism is justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problems in the discrete W 2.(2.1) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original quasilinear parabolic problem is proved.

关键词： difference scheme, intrinsic parallelism,quasilinear parabolic system, convergence, stability.

DIFFERENCE SCHEMES WITH NONUNIFORM MESHES FOR NONLINEAR PARABOLIC SYSTEM

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Journal of computational mathematics 1996年第4期14卷 319-335页

作者： Y.L. Zhou(laboratory of computational {2., Centre for Nonlinear Studies, institute of applied{2. and computational mathematics, beijing, china) 北京应用物理和计算数学所北京

The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general difference schemes with unequal meshsteps are established by the fixed point technique. The absolute and relative convergence of the discrete vector solution are justified by a series of a priori estimates. The analysis of mentioned problems are based on the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem of the nonlinear parabolic system.

关键词： Math QT DIFFERENCE SCHEMES WITH NONUNIFORM MESHES FOR NONLINEAR PARABOLIC SYSTEM

Mathematical stencil and its application in finite difference approximation to the Poisson equation

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Science china mathematics 2005年第10期48卷 1421-1429页

作者： FENG Hui~1 ZHANG Baolin~(1,2. & LIU Yang~1 1. Department of mathematics,Wuhan University,Wuhan 430072.china {2.,institute of applied physics and computational mathematics,beijing 1.0088. china Department of mathematics Wuhan University Wuhan China laboratory of computational {2. Institute of Applied Physics and Computational Mathematics Beijing China

The concept of mathematical stencil and the strategy of stencil elimination for solving the finite difference equation is presented, and then a new type of the iteration algorithm is established for the Poisson equation. The new algorithm has not only the obvious property of parallelism, but also faster convergence rate than that of the classical Jacobi iteration. Numerical experiments show that the time for the new algorithm is less than that of Jacobi and Gauss-Seidel methods to obtain the same precision, and the computational velocity increases obviously when the new iterative method, instead of Jacobi method, is applied to polish operation in multi-grid method, furthermore, the polynomial acceleration method is still applicable to the new iterative method.

关键词： mathematical stencil, stencil elimination, Poisson equation, finite difference, iterative algorithm,parallelism.

Cluster identification and characterization of physical fields

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Science china(physics,Mechanics & Astronomy) 2010年第9期53卷 1610-1618页

作者： ZHANG GuangCai,XU AiGuo,LU Guo & MO ZeYao National Key laboratory of computational {2.,institute of applied physics and computational mathematics,beijing 1.0088.china National Key laboratory of computational {2. Institute of Applied Physics and Computational Mathematics Beijing China

The description of complex configuration is a difficult *** present a powerful technique for cluster identification and *** scheme is designed to treat and analyze the experimental and/or simulation data from various *** main steps are as *** first divide the space using face or volume elements from discrete ***,we combine the elements with the same and/or similar properties to construct clusters with special physical *** the algorithm,we adopt an administrative structure of a hierarchy-tree for spatial bodies such as points,lines,faces,blocks,and *** fast search algorithms with the complexity lnN are *** establishment of the hierarchy-tree and the fast searching of spatial bodies are general,which are independent of spatial ***,it is easy to extend the method to other *** a verification and validation,we applied this method and analyzed some two-dimensional and three-dimensional random data.

关键词： spatial hierarchical tree fast search complex configuration dynamic physical fields cluster identification

GENERAL INTERPOLATION FORMULAS FOR SPACES OF DISCRETE FUNCTIONS WITH NONUNIFORM MESHES

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Journal of computational mathematics 1995年第1期13卷 70-93页

作者： Zhou Yu-lin(institute of applied physics and computational mathematics, beijing, {2. ) 北京应用物理和计算数学所北京

The unequal meshsteps are unavoidable in general for scientific and engineering computations especially in large Scale computations. The analysis of difference schemes with nonuniform meshes is very rare even by use of fully heuristic methods. For the purpose of the systematic and theoretical study of the finite difference method with nonuniform meshes for the problems of partial differential equations, the general interpolation formulas for the spaces of discrete functions of one index with unequal meshsteps are established in the present work. These formulas give the connected relationships among the norms of various types, such as' the sum of powers of discrete values, the discrete maximum modulo, the discrete Holder and Lipschitz coefficients.

关键词： Discrete functions Interpolation Mathematical constants Partial differential equations Difference quotients Real lines Finite difference methods Mathematical theorems Ratios Engineering