小波分析在流体方程中的应用研究
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摘要
小波分析则是Fourier分析的发展和完善,小波分析也是一门新兴理论,它被广泛地应用于各个领域。作为80年代末期出现的时频分析工具,小波变换在信号与图像处理等领域里己经得到了成功的应用,并凭借其自身的诸多优点成为了JPEG2000的标准。由于小波分析的发展是以解决实际问题应用为出发点,而后上升到辐射多学科的理论,所以小波分析一次又一次形成研究热潮,成为国际研究热点。
求解动力学偏微分方程特别是非线性动力学偏微分方程对力学的发展起到了非常重要的作用。目前求解非线性偏微分方程的方法主要是进行数值求解基于多尺度分析的尺度函数和小波函数很好的分析特性和计算特性,充分利用这些特性以小波作为基函数离散微分方程,则我们可以得到的代数方程。
小波分板在流体方面的数值计算不仅湍流数值模拟中的应用少,没有三维数值模拟,层流方面也还是主要集中在一维问题上,代表性的数值计算便是Burgers方程,本文的目的方法成果结论新见解如下;
目的;
使小波在一维方面优异的计算结果进入流体领域高维问题。
方法;
采用张量积小波,时间离散采用Euler法,Von Neumann稳定性分析,迭代矩阵的收敛问题及矩阵病态程度分析。
成果;
完成一二三维的各种流体方程小波算例及各种分析算例9个,形成二三维线性稳定性为主的理论框架,完成一二三维保障数值计算稳定的分析,小波矩阵的迭代收敛及矩阵病态程度的分析研究。
结论;
小波技术对于流体高维方程是有效的和先进的,小波技术对于定常NS方程组是有效的和先进的。二三维线性稳定性理论说明可以建立小波技术求解流体方程的理论体系。一二三维保障数值计算稳定的分析为工程上的应用奠定了基础。
新见解;
与差分方法的比较展现了小波技术的巨大潜力。随着小波技术的深入,小波方法很可能是求解偏微分方程发展最快,最有前景,最强有力的技术之一。
The wavelet analysis is the development and perfection of the Fourier analysis.Wavelet analysis theory is also a arisen science which was applied extensively toevery domain. Being a time-frequency analysis tool in 1980s', Wavelet transform hassucceeded to be applied to the signal and image processing domain and been thecriterion of JPEG 2000 by it's advantages Since the development of the waveletanalysis is the basis to solve some practical problems, and then, it develops into aradioactive multi-disciplined theory, now it has become a hot field in the researchinternationally. The study of solving partial differential equations has played animportant role in the development of mechanics. At present, One of the primarymethods to solve partial differential equations is numerical method. based onMultiresolution Analysis, scale functions and wavelet functions have good analysisand computation characteristic which are been made the best of to discrete thedifferential equations. Then some algebra equations can be acquired. The applicationof wavelet analysis in Numerical Simulation of turbulence is No 3-D Numericalsimulation of turbulent flowfield. The research for the laminar flows has focused onits 1 -D numerical Simulation. Then representative numerical Simulations are solvingBurgers differential equation The purpose, method, results, conclusions and new viewof this paper is as follow:
The purpose of this paper:
Make the excellent numerical Simulation result enter into multivariate fluidcalculation.
The method of this paper:
We use wavelet bases formed by tensor products, Euler method was used in timediscretization. This paper researches the problem using the Von Neumann method.We study the convergence property of iteration matrices generated by the waveletmethod and Study on ill-conditioned problems of matrices generated by the waveletmethod.
The results of this paper: carrying out the nine numerical Simulations for fluid Equation, the stability theory is mainly by 2-D and 3-D linear stability theory has been formed. Carrying out 1-D , 2-D and 3-D Stability Analysis of iteration matrices and ill-conditioned problems of matrices for the safeguard of numerical calculation.
The conclusions of this paper:
The wavelet analysis is an advanced and valuable numerical simulation method for multivariate fluid calculation and for Steady Navier-Stokes Equations. The task of constructing linear stability theory can be realized. 1-D, 2-D and 3-D Stability Analysis for the safeguard of numerical calculation forms the base of the application of fluid numerical simulation.
The new view of this paper:
The wavelet theory shows its potential in fluid numerical simulation when it was compared with finite-difference method. With the in-depth development of the wavelet theory, the wavelet analysis may probably be one of fastest-growing, the most prospect and the most powerful numerical simulation technique.
求解动力学偏微分方程特别是非线性动力学偏微分方程对力学的发展起到了非常重要的作用。目前求解非线性偏微分方程的方法主要是进行数值求解基于多尺度分析的尺度函数和小波函数很好的分析特性和计算特性,充分利用这些特性以小波作为基函数离散微分方程,则我们可以得到的代数方程。
小波分板在流体方面的数值计算不仅湍流数值模拟中的应用少,没有三维数值模拟,层流方面也还是主要集中在一维问题上,代表性的数值计算便是Burgers方程,本文的目的方法成果结论新见解如下;
目的;
使小波在一维方面优异的计算结果进入流体领域高维问题。
方法;
采用张量积小波,时间离散采用Euler法,Von Neumann稳定性分析,迭代矩阵的收敛问题及矩阵病态程度分析。
成果;
完成一二三维的各种流体方程小波算例及各种分析算例9个,形成二三维线性稳定性为主的理论框架,完成一二三维保障数值计算稳定的分析,小波矩阵的迭代收敛及矩阵病态程度的分析研究。
结论;
小波技术对于流体高维方程是有效的和先进的,小波技术对于定常NS方程组是有效的和先进的。二三维线性稳定性理论说明可以建立小波技术求解流体方程的理论体系。一二三维保障数值计算稳定的分析为工程上的应用奠定了基础。
新见解;
与差分方法的比较展现了小波技术的巨大潜力。随着小波技术的深入,小波方法很可能是求解偏微分方程发展最快,最有前景,最强有力的技术之一。
The wavelet analysis is the development and perfection of the Fourier analysis.Wavelet analysis theory is also a arisen science which was applied extensively toevery domain. Being a time-frequency analysis tool in 1980s', Wavelet transform hassucceeded to be applied to the signal and image processing domain and been thecriterion of JPEG 2000 by it's advantages Since the development of the waveletanalysis is the basis to solve some practical problems, and then, it develops into aradioactive multi-disciplined theory, now it has become a hot field in the researchinternationally. The study of solving partial differential equations has played animportant role in the development of mechanics. At present, One of the primarymethods to solve partial differential equations is numerical method. based onMultiresolution Analysis, scale functions and wavelet functions have good analysisand computation characteristic which are been made the best of to discrete thedifferential equations. Then some algebra equations can be acquired. The applicationof wavelet analysis in Numerical Simulation of turbulence is No 3-D Numericalsimulation of turbulent flowfield. The research for the laminar flows has focused onits 1 -D numerical Simulation. Then representative numerical Simulations are solvingBurgers differential equation The purpose, method, results, conclusions and new viewof this paper is as follow:
The purpose of this paper:
Make the excellent numerical Simulation result enter into multivariate fluidcalculation.
The method of this paper:
We use wavelet bases formed by tensor products, Euler method was used in timediscretization. This paper researches the problem using the Von Neumann method.We study the convergence property of iteration matrices generated by the waveletmethod and Study on ill-conditioned problems of matrices generated by the waveletmethod.
The results of this paper: carrying out the nine numerical Simulations for fluid Equation, the stability theory is mainly by 2-D and 3-D linear stability theory has been formed. Carrying out 1-D , 2-D and 3-D Stability Analysis of iteration matrices and ill-conditioned problems of matrices for the safeguard of numerical calculation.
The conclusions of this paper:
The wavelet analysis is an advanced and valuable numerical simulation method for multivariate fluid calculation and for Steady Navier-Stokes Equations. The task of constructing linear stability theory can be realized. 1-D, 2-D and 3-D Stability Analysis for the safeguard of numerical calculation forms the base of the application of fluid numerical simulation.
The new view of this paper:
The wavelet theory shows its potential in fluid numerical simulation when it was compared with finite-difference method. With the in-depth development of the wavelet theory, the wavelet analysis may probably be one of fastest-growing, the most prospect and the most powerful numerical simulation technique.
引文
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