In this study, a classical spectral-finite difference scheme (SFDS) for the three-dimensional (3D) parabolic equation is reduced by using proper orthogonal decomposition (POD) and singular value decomposition (SVD). First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second, the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reducedorder extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine the classical finite volume element (FVE) method with the POD method to obtain a reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional solute transport problems, which have real life practical applications. We then provide error estimates between the reduced-order POD FVE solutions and classical FVE solutions and we provide implementation of an extrapolation algorithm for solving the reduced-order FVE formulation. Thus, we provide the theoretical basis for practical applications. A numerical example is then used to ascertain that the results of numerical computation are consistent with the theoretical derivations. Moreover, it is shown that the reduced-order FVE formulation based on POD method is both feasible and efficient for solving two-dimensional solute transport problems.

In this paper, we extend the applications of proper orthogonal decomposition (POD) method, i.e., apply POD method to a mixed finite element (MFE) formulation naturally satisfied Brezz-Babuska for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.

At first, a semi-discrete formulation with respect to time for the non-stationary conduction–convection problem is recalled. Then, a fully discrete stabilized mixed finite volume element (SMFVE) formulation based on two local Gauss integrals and parameter-free is established directly from the semi-discrete formulation with respect to time for the non-stationary conduction–convection problem. Following this, the error estimates for the fully discrete SMFVE solutions are derived by means of the standard mixed finite element method. Finally, some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results, the degrees of freedom of the fully discrete SMFVE formulation are far fewer than those of the classical finite volume element (FVE) formulation without any stabilization, and its numerical solutions are more stable than those of the classical FVE formulation without any stabilization, thus validating that the fully discrete SMFVE formulation is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem.

本文用分裂正定混合有限元方法研究二阶粘弹性方程.首先构造一种新的分裂正定混合变分形式和基于这种分裂正定混合变分形式关于时间的半离散格式,然后绕开关于空间变量的半离散化格式,直接从时间半离散出发构造出全离散化的分裂正定混合有限元格式,并给出这种分裂正定混合有限元解的误差估计.这种研究思路使得理论论证变得更简单,这是处理二阶粘弹性方程的一种新的尝试.

In this paper,a traffic flow Aw-Rascle-Zhang(ARZ) model is studied with a proper orthogonal decomposition(POD) technique.A extrapolation reduced-order finite difference scheme(FDS) based on POD method with lower dimension is established.And a numerical example is used to verify that the results of numerical computation are consistent with theoretical conclusions.Moreover,it is shown that the extrapolation reduced-order FDS based on POD method is feasible and efficient for finding numerical solutions for traffic flow equation.

In this paper, a proper orthogonal decomposition (POD) method is used to deal with a classical Crank-Nicolson finite volume element (CNFVE) method for two-dimensional parabolic equations. A reduced-order CNFVE formulation with lower dimensions and sufficiently high accuracy based on POD technique is established, the error estimates between reduced-order CNFVE solutions based on the POD method and classical CNFVE solutions are provided, and the extrapolation algorithm for solving reduced-order CNFVE formulation is implemented. Some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions. Moreover, it is shown that the reduced-order CNFVE formulation based on POD method is feasible and efficient for solving two-dimensional parabolic equations.

提出了一种基于形态学结构运算操作的SAR图像道路检测方法。该方法首先进行滤波去噪处理,经过实验对比分析得出各种滤波的优劣,再利用形态学操作对原始图像进行高低帽图像增强处理,运用线性生长和区域填充方法进行线性提取,然后利用边缘检测算法检测目标,利用连通域目标的面积和空间关系等特征去除少量误提道路特征,最后按照一定规则利用霍夫变换进行自动道路检测。实验表明,新方法可以取得很好的效果。

高等教育系统要大力推进和谐校园建设来夯实社会主义和谐社会,该文阐述了为了构建和谐校园,在离散数学教学中努力抓好素质教育",以学生发展为宗旨"的教学理念的实现。

In this paper, a classical fully second-order finite difference scheme (FDS) for non-stationary Burgers equation is reduced with a proper orthogonal decomposition method and singular value decomposition technique. A reduced-order FDS of second-order accuracy about time and spacial variables is derived. The error estimates of the reduced-order FDS solutions and the implementation of its extrapolation algorithm are provided. Finally, a numerical example illustrates the fact that the results of numerical computation are consistent with theoretical conclusions.

The non-stationary conduction–convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element (FVE) method. A fully discrete FVE formulation and the error estimates between the fully discrete FVE solutions and the accuracy solution are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the FVE formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non-stationary conduction–convection problem.

本文研究二维Sobolev方程的有限体积元方法,给出一种全离散化有限体积元格式及其有限体积元解的误差估计,并用数值例子说明数值计算的结果与理论结果是相吻合的,进一步说明了有限体积元方法比其他数值方法更优越.

In this paper, a proper orthogonal decomposition (POD) technique is used to establish a reduced-order finite difference (FD) extrapolation algorithm with lower dimensions and sufficiently high accuracy for the non-stationary Navier–Stokes equations, and the error estimates between the reduced-order FD solutions and the classical FD solutions and the implementation for solving the reduced-order FD extrapolation algorithm are provided. Two numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order FD extrapolation algorithm based on POD method is feasible and efficient for solving the non-stationary Navier–Stokes equations.

In this paper,a finite volume element(FVE) formulation for two-dimension solute transport problems is derived,error estimates between FVE solutions and accurate solutions are provided.It is shown by numerical examples that FVE formulation is stabler that finite element formulation for solving two-dimension solute transport problems,

In this paper,two-dimensional parabolic equations are studied with mixed method. A type of new mixed variational formulations,time-semi-discrete mixed equations,and fully discrete mixed finite element formulations are derived,where Brezzi-Babuska's condition is automatically satisfied.And some rigorous error analyses are provided.The degrees of freedom are not only minimum for these mixed finite element formulations,but their error estimates also optimal order.Moreover,it is shown that the methods here are the improvements and renovations for existing all methods.

A proper orthogonal decomposition(POD) method is applied to a usual second order time accurate finite element(SOTAFE) formulation of time second order central difference for parabolic equations so that it is reduced into a SOTAFE formulation of time second order central difference with fewer degrees of freedom.The errors between the reduced SOTAFE solutions of time second order central difference based on POD approach and the usual SOTAFE solutions of time second order central difference are analyzed.Numerical examples show that the reduced SOTAFE formulation of time second order central difference based POD approach can save a lot of degrees of freedom in a way that guarantees a sufficiently small errors between the reduced SOTAFE solutions of time second order central difference based on POD approach and the usual SOTAFE solutions of time second order central difference.Moreover,this verifies the reduced SOTAFE formulation of time order central difference based on POD approach is feasible and efficient solving parabolic equations.

This article constructs a manageable and expandable mountain model algorithm by making use of a graphic primitive of OpenGL and brings about visualization of mountain from single mountain to many mountains combined with color rendering,texture mapping and light processing.The model not only can render a common mountain,but also can be used as the one that is similar to mountain,such as stalactites and mountain slope.Besides,this model in application can be used in real-time rendering CAD modeling system and other three-dimensional scene modeling system.

In this paper,a stabilized second order mixed finite element formulation based on bubble functions for plane elasticity problems is studied,and a simplified second order mixed finite element formulation with less freedom degrees is obtained by eliminating all bubble functions.It is shown by analyzing error that the convergence of the simplified and stabilized second order mixed finite element formulation eliminated all bubble functions is the same as that of the stabilized second order mixed finite element formulation with bubble functions,but it can save 18N_P freedom degrees(where N_p is the number of vertices of triangularization).

In this paper,a finite volume element method for non-stationary Stokes equation is studied and a stabilized fully discrete finite volume element formulation based on on two local Gauss integrals for non-stationary Stokes equation is derived.The errors of solution for this formulation is analyzed.

A proper orthogonal decomposition (POD) method is applied to a usual second-order time accurate Crank-Nicolson finite element (CNFE) formulation for parabolic equations such that it is reduced into a second- order time accurate CNFE formulation with fewer degrees of freedom and high enough accuracy. The errors between the reduced second-order time accurate CNFE solutions and the usual second-order time accurate CNFE solutions are analyzed. It is shown by numerical examples that the reduced second-order time accurate CNFE formulation can greatly save degrees of freedom in a way that guarantees a sufficiently small errors between the reduced second-order time accurate CNFE solutions and the usual second-order time accurate CNFE solutions. The time step of the reduced second-order time accurate CNFE formulation is ten times that of the first-order time accurate reduced finite element formulation such that it could obtain very quickly the numerical solution at the moment wanted, alleviate the computer truncation error, and improve rate and accuracy in the computational process. Moreover, it is also shown that the reduced second-order time accurate CNFE formulation is feasible and efficient solving parabolic equations.