流体力学中几类波方程的有限体积元方法
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摘要
有限体积元方法是数值求解偏微分方程的一类重要的数值工具.由于该方法易于执行、剖分灵活,并且能够自然保持主要物理守恒律,它越来越受到研究者的重视.本文主要研究了流体力学中几类波动方程的有限体积元方法.针对不同的问题构造了相应的有限体积元格式,进一步对微分方程进行了数值研究.
首先考察带延迟项的双曲偏微分-差分方程.对该方程设计了有限体积元格式及迎风有限体积元格式,并给出迎风有限体积元格式的L2误差估计.数值试验验证了格式的有效性及收敛性.
其次研究带有随机效应的衰减改进Boussinesq方程.对于随机衰减改进Boussinesq方程,空间方向用二次Lagrange函数逼近,时间方向用三阶强稳定格式近似,随机项用蒙特卡罗方法离散,构造了全离散的有限体积元格式.利用得到的格式,数值研究了随机效应对系统质量及孤立波振幅的影响.
接下来讨论定义在球面上的二维准地转方程.对于全球准地转方程提出了Fourier有限体积元方法.纬度方向用一次多项式近似,经度方向用Fourier基逼近,时间方向用蛙跳格式离散,建立了Fourier有限体积元格式.数值结果表明该格式具有二阶收敛,能够保持能量和涡度拟能守恒,而且能克服极点问题.
最后研究了基于无导数优化算法的空气质量最优控制问题.应用特征有限差分方法数值求解描述污染物运动发展的空气污染模型,以排放污染物的工厂位置为决策变量,定义相关的目标函数,对所构造的极小化问题使用无导数优化进行求解.一系列的数值试验表明,无导数算法能够对我们构造的问题进行求解,经优化过的工厂位置能够以较低的成本,满足空气质量指标.
The finite volume element method has been an important numerical tool for solv-ing partial differential equations. Since the method is easily implemented, can effi-ciently deal with complex geometries and naturally preserve main physical conserva-tion laws, it has attracted considerable interest. This work is devoted to finite volume element methods for some wave equations in fluid dynamics. We design some effec-tive finite volume element schemes to study these wave equations.
Firstly, we consider the hyperbolic partial differential-difference equation with shift and design standard finite volume scheme and upwind finite volume element scheme for solving it. At the same time, L2error estimate is given for upwind finite volume element scheme. Several numerical experiments are performed to check the efficiency and convergence of the numerical schemes.
Secondly, we study the stochastic damped improved Boussinesq equation. To numerically solve the equation, we use quadratic Lagrange functions to approximate space derivative, three-order strong-stability-preserving scheme to discretize time derivative and Monte Carlo method to deal with stochastic term. We have obtained fully discrete finite volume element scheme for the equation. By the scheme, we nu-merical investigate the influence of a noise term on mass of system and amplitude of solitary wave.
Next, we consider the two-dimensional quasi-geostrophic equations on a sphere and propose a new Fourier finite volume element method. By using piecewise linear functions in the latitudinal direction, Fourier discretization in the longitudinal direc-tion and leap-frog scheme to discretize time derivative, we can get the Fourier finite volume element schemes. Some numerical results illustrate that the new method is second-order convergence, and can conserve the energy and enstrophy of system and overcome pole problem.
At last, we discuss optimal control of air quality based on derivative-free opti-mization method. The characteristic finite difference method is used to solving the air pollution model which describe the development of pollutant. The locations of industrial pollutant sources are chosen as the decision variables. By defining rel- ative objective functions, we solve the optimization problems using derivative-free optimization method. Some numerical examples demonstrate that derivative-free op-timization method can be used to solve these problems. The optimal locations of pollution sources can meet the air quality index at low cost.
首先考察带延迟项的双曲偏微分-差分方程.对该方程设计了有限体积元格式及迎风有限体积元格式,并给出迎风有限体积元格式的L2误差估计.数值试验验证了格式的有效性及收敛性.
其次研究带有随机效应的衰减改进Boussinesq方程.对于随机衰减改进Boussinesq方程,空间方向用二次Lagrange函数逼近,时间方向用三阶强稳定格式近似,随机项用蒙特卡罗方法离散,构造了全离散的有限体积元格式.利用得到的格式,数值研究了随机效应对系统质量及孤立波振幅的影响.
接下来讨论定义在球面上的二维准地转方程.对于全球准地转方程提出了Fourier有限体积元方法.纬度方向用一次多项式近似,经度方向用Fourier基逼近,时间方向用蛙跳格式离散,建立了Fourier有限体积元格式.数值结果表明该格式具有二阶收敛,能够保持能量和涡度拟能守恒,而且能克服极点问题.
最后研究了基于无导数优化算法的空气质量最优控制问题.应用特征有限差分方法数值求解描述污染物运动发展的空气污染模型,以排放污染物的工厂位置为决策变量,定义相关的目标函数,对所构造的极小化问题使用无导数优化进行求解.一系列的数值试验表明,无导数算法能够对我们构造的问题进行求解,经优化过的工厂位置能够以较低的成本,满足空气质量指标.
The finite volume element method has been an important numerical tool for solv-ing partial differential equations. Since the method is easily implemented, can effi-ciently deal with complex geometries and naturally preserve main physical conserva-tion laws, it has attracted considerable interest. This work is devoted to finite volume element methods for some wave equations in fluid dynamics. We design some effec-tive finite volume element schemes to study these wave equations.
Firstly, we consider the hyperbolic partial differential-difference equation with shift and design standard finite volume scheme and upwind finite volume element scheme for solving it. At the same time, L2error estimate is given for upwind finite volume element scheme. Several numerical experiments are performed to check the efficiency and convergence of the numerical schemes.
Secondly, we study the stochastic damped improved Boussinesq equation. To numerically solve the equation, we use quadratic Lagrange functions to approximate space derivative, three-order strong-stability-preserving scheme to discretize time derivative and Monte Carlo method to deal with stochastic term. We have obtained fully discrete finite volume element scheme for the equation. By the scheme, we nu-merical investigate the influence of a noise term on mass of system and amplitude of solitary wave.
Next, we consider the two-dimensional quasi-geostrophic equations on a sphere and propose a new Fourier finite volume element method. By using piecewise linear functions in the latitudinal direction, Fourier discretization in the longitudinal direc-tion and leap-frog scheme to discretize time derivative, we can get the Fourier finite volume element schemes. Some numerical results illustrate that the new method is second-order convergence, and can conserve the energy and enstrophy of system and overcome pole problem.
At last, we discuss optimal control of air quality based on derivative-free opti-mization method. The characteristic finite difference method is used to solving the air pollution model which describe the development of pollutant. The locations of industrial pollutant sources are chosen as the decision variables. By defining rel- ative objective functions, we solve the optimization problems using derivative-free optimization method. Some numerical examples demonstrate that derivative-free op-timization method can be used to solve these problems. The optimal locations of pollution sources can meet the air quality index at low cost.
引文
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[3]李荣华.两点边值问题的广义差分法.吉林大学自然科学学报,1(1982)26-40.
[4]R. Li. Generalized difference methods for a nonlinear Dirichlet problem. SIAM J. Numer. Anal.,24(1987) 77-88.
[5]Z. Cai, J. Mandel, and S. Mccormick. The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal.,28(1991) 392-402.
[6]R. Li, Z. Chen, and W. Wu. Generalized difference methods for differential equation:numerical analysis of finite volume methods. New York:Marcel Dekker,1999.
[7]R. Ewing, T. Lin, and Y. Lin. On the accuracy of the finite volume ele-ment method based on piecewise linear polynomials. SIAM J. Numer. Anal., 39(2002)1865-1888.
[8]J. Xu and Q. Zou. Analysis of linear and quadratic simplicial finite volume methods for elliptic equations. Numer. Math.,111(2009) 469-492.
[9]Z. Zhang. Error estimate of finite volume element method for the pollution in groundwater flow. Numer. Meth. Part. D. E.,25(2009) 259-274.
[10]P. Wang and Z. Zhang. Quadratic finite volume element method for the air pollution model. Int. J. Comput. Math.,87(2010) 2925-2944.
[11]Z. Zhang and F. Lu. Quadratic finite volume element method for the improved Boussinesq equation. J. Math. Phys.,53(2012) 013505.
[12]P. Chatzipantelidis. A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions. Numer. Math.,82(1999) 409-432.
[13]C. Bi and L. Li. Mortar finite volume method with Adini element for bihar-monic problem. J. Comput. Math.,22(2004) 475-488.
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