Sobolev方程基于POD的降阶外推差分算法

用奇异值分解和特征投影分解(proper orthogonal decomposition, 简记POD)方法建立Sobolev方程的一种降阶外推有限差分算法, 并给出误差估计.最后用数值例子,验证基于POD方法降阶外推有限差分算法的可行性和有效性.     

二维双曲方程基于POD方法的降阶有限差分外推迭代格式

利用特征投影分解(POD)方法建立二维双曲型方程的一种基于POD方法的含有很少自由度但具有足够高精度的降阶有限差分外推迭代格式,给出其基于POD方法的降阶有限差分解的误差估计及基于POD方法的降阶有限差分外推迭代格式的算法实现.用一个数值例子去说明数值计算结果与理论结果相吻合.进一步说明这种基于POD方法的降阶有限差分外推迭代格式对于求解二维双曲方程是可行和有效的.     

二维土壤溶质输运方程基于POD方法的降阶CN有限体积元外推算法

利用Crank-Nicolson(CN)有限体积元方法和特征投影分解方法建立二维土壤溶质输运方程的一种维数很低、精度足够高的降阶CN有限体积元外推算法,并给出这种外推算法的降阶CN有限体积元解的误差估计和算法的实现.最后用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶CN有限体积元外推算法的优越性.     

非定常Burgers方程基于POD方法的全二阶差分格式降阶外推算法

利用特征投影分解方法和奇值分解技术对非定常Burgers 方程的经典全二阶有限差分格式做降阶处理, 给出一种时间和空间变量都是二阶精度的降阶差分格式, 并给出这种降阶全二阶精度差分解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的.     

守恒高阶各向异性交通流模型基于POD方法的降阶外推差分格式

利用Godunov流方法和特征投影分解方法,对守恒高阶各向异性交通流模型建立一种自由度很少、精度足够高的降阶外推差分算法, 并给出这种降阶外推差分算法近似解的误差估计和算法实现.最后,用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶外推差分算法的优越性.     

二维土壤溶质输运方程基于POD方法的降阶CN有限元外推算法

首先给出二维土壤溶质输运方程时间二阶精度的Crank-Nicolson(CN)时间半离散化格式和时间二阶精度的全离散化CN有限元格式及其误差分析.然后利用特征投影分解(proper orthogonal decomposition,简记为POD)方法对二维土壤溶质输运方程的经典CN有限元格式做降阶处理,建立一种具有足够高精度、自由度很少的降阶CN有限元外推格式,并给出这种降阶CN有限元解的误差估计和外推算法的实现.最后用数值例子说明数值结果与理论结果是相吻合的.     

结构三角网上抛物方程的有限差分区域分解算法

1引言对于大型科学与工程计算问题,并行计算是必需的．构造高效率的数值并行方法一直是人们关心的问题,并且已有了大量的研究．在三层交替计算方法的研究中出现了许多既具有明显并行性又绝对稳定的差分格式(见[1]-[5])．在只涉及两个时间层的算法研究中,Dawson等人(见[6])首先发展了求解一维热传导方程的区域分解算法,并将其推广到     

热传导对流方程基于POD的差分格式

本文用奇值分解和特征投影分解(proper orthogonal decomposition,简记为POD)研究热传导对流方程,导出其基于POD的一种简化的差分格式,并分析通常的差分格式的解和基于POD的简化的差分格式的解之间的误差估计.最后用方腔流数值例子验证本文的理论的正确性,从而验证了用基于POD的简化的差分格式解热传导对流方程的有效性.     

非饱和土壤水流问题基于POD 方法的降阶有限体积元格式及外推算法实现

利用特征投影分解(proper orthogonal decomposition, 简记为POD) 方法对非饱和土壤水流问题的经典有限体积元格式做降阶处理, 建立一种具有足够高精度维数较低的降阶有限体积元格式, 并给出这种降阶有限体积元解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的. 进一步表明了基于POD 方法的降阶有限体积元格式对求解非饱和土壤水流问题数值解是可靠和有效的.     

一类无结构三角网上抛物方程的有限差分区域分解算法

本文讨论了一类在无结构三角网上数值求解二维热传导方程的有限差分区域分解算法．在这个算法中,将通过引进两类不同类型的内界点,将求解区域分裂成若干子区域．一旦内界点处的值被计算出来,其余子区域上的计算可完全并行．本文得到了稳定性条件和最大模误差估计,它表明我们的格式有令人满意的稳定性和较高的收敛阶．     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.     

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

区域分裂法求非线性抛物方程的扩张混合元解

针对非线性抛物方程,给出了全离散的扩张混合元格式,利用一个建立在非重叠型区域分裂技巧上的并行迭代法求解了最后的非线性代数方程组,证明了迭代法的收敛性并给出了最优阶的误差估计.     

A Locally Conservative Eulerian-Lagrangian Finite Difference Method for a Parabolic Equation

The object of this paper is to define a finite difference analogue of a locally conservative Eulerian—Lagrangian method based on mixed finite elements and to prove its convergence. The method is appropriate for convection-dominated diffusive processes; here, it will be considered in the case of a semilinear parabolic equation in a single space variable.This revised version was published online in October 2005 with corrections to the Cover Date.     

Finite element formulation based on proper orthogonal decomposition for parabolic equations

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method. This work was supported by National Natural Science Foundation of China (Grant Nos. 10871022, 10771065, and 60573158) and Natural Science Foundation of Hebei Province (Grant No. A2007001027)     

解三维抛物型方程的一个高精度显式差分格式

提出了求解三维抛物型方程的一个高精度显式差分格式.首先,推导了一个特殊节点处一阶偏导数(■u)/(■/t)的一个差分近似表达式,利用待定系数法构造了一个显式差分格式,通过选取适当的参数使格式的截断误差在空间层上达到了四阶精度和在时间层上达到了三阶精度.然后,利用Fourier分析法证明了当r1/6时,差分格式是稳定的.最后,通过数值试验比较了差分格式的解与精确解的区别,结果说明了差分格式的有效性.     

交通流模型基于特征投影分解技术的外推降维有限差分格式

利用特征正交分解(proper orthogonal decomposition,简记为POD)技术研究交通流的Aw-Rascle-Zhang(ARZ)模型. 建立一种基于 POD方法维数较低的外推降维有限差分格式, 并用数值例子检验数值计算结果与理论结果相吻合, 进一步表明基于POD方法的外推降维有限差分格式对于求解交通流方程数值解是可行和有效的.