Sobolev方程的全离散有限体积元格式及数值模拟

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.     

Sine-Gordon方程的混合有限体积元方法及数值模拟

研究了在Dirichlet边界条件和Neumann边界条件下一维sine-Gordon方程的混合有限体积元方法.通过引入将试探函数空间映射到检验函数空间的迁移算子γh,结合混合有限元方法和有限体积元方法,构造了半离散格式,时间显式和隐式全离散混合有限体积元格式.给出了显格式离散解的稳定性分析,并得到了三种格式的最优阶误差估计.最后,给出数值算例来验证理论分析结果和数值格式的有效性.     

二维非饱和土壤水分运动问题的半离散有限体积元模拟

本文利用基于重心对偶剖分的有限体积元法建立了二维非饱和土壤水分运动问题的数值逼近格式,讨论了离散有限体积元解的存在唯一性,并给出了最优误差估计的证明.最后给出数值算例,模拟结果表明,利用有限体积元格式来求解二维非饱和土壤水分运动问题是可靠的,且该格式具有稳定性和可实用性.     

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.     

非定常Stokes方程的稳定化全离散有限体积元格式

本文研究非定常Stokes方程的有限体积元方法,给出一种基于两个局部高斯积分的稳定化全离散格式,并给其有限体积元解的误差分析.     

二维非线性Schr(o)dinger方程的辛与多辛格式

对满足周期边界条件的二维非线性Schrödinger方程,运用中心差分对该方程进行空间离散, 得到一个有限维Hamilton系统,然后用隐式Euler中点格式进行时间离散得到其辛格式. 针对该方程的多辛形式, 运用有限体积法离散,得到一种直平行六面体上的中点型多辛格式. 用所构造的辛与多辛格式对二维非线性Schrödinger方程的平面波解和奇异解进行数值模拟,结果验证了所构 造格式的有效性. 最后, 根据计算结果,对两种格式进行了分析和比较.       

非线性对流扩散方程沿特征线的多步有限体积元格式

对于二维非线性对流扩散方程构造了沿特征线方向的多步有限体积元格式．关于空间采用二次有限体积元方法离散，关于时间采用多步法进行离散，获得了O(Δt^2 h^2)形式的误差估计．本文最后给出的数值算例表明了方法的有效性．     

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.     

非定常 N-S 方程的非线性 Galerkin 算法及其误差估计

本文给出了二维非定常N-S方程的三种数值格式,其中空间变量用谱非线性Galerkin算法进行离散,时间变量用有限差分离散,并研究了这些格式数值解的逼近精度．最后,给出了部分数值计算结果．     

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.     

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.     

A finite volume element formulation and error analysis for the non-stationary conduction–convection problem

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.     

二维土壤溶质输运方程的有限体积元格式

本文导出二维的土壤溶质输运方程的有限体积元格式, 并分析其误差.通过数值例子说明, 有限体积元格式比有限元格式稳定.     

A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.     

A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem

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 a classical finite volume element (FVE) method with POD method to establish a reduced FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional viscoelastic problem with real practical applied background, and analyze the errors between the reduced POD FVE solution and the classical FVE solution so as to provide scientific theoretic basis for service applications. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced FVE formulation based on POD method is feasible and efficient for solving two-dimensional viscoelastic problem.     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

Numerical solution of a multidimensional sedimentation problem using finite volume-element methods

We are interested in the reliable simulation of the sedimentation of monodisperse suspensions under the influence of body forces. At the macroscopic level, the complex interaction between the immiscible fluid and the sedimentation of a compressible phase may be governed by the Navier–Stokes equations coupled to a nonlinear advection–diffusion–reaction equation for the local solids concentration. A versatile and effective finite volume element (FVE) scheme is proposed, whose formulation relies on a stabilized finite element (FE) method with continuous piecewise linear approximation for velocity, pressure and concentration. Some numerical simulations in two and three spatial dimensions illustrate the features of the present FVE method, suggesting their applicability in a wide range of problems.