对流扩散方程的新型Crank-Nicholson差分格式

本文针对一维非定常对流扩散方程，构造了一种对角元严格占优的Crank-Nicholson差分格式，利用能量估计的方法对该格式做了稳定性分析．收敛性收分析以及误差估计．数值试验结果表明．该格式具有良好的稳定性．     

一种求解运动曲面上对流扩散方程的三维水平集方法

给出了一种求解运动曲面上对流扩散方程的三维水平集算法. 水平集函数被用来表示曲面.曲面上的微分方程及其解通过水平集方法被延拓到包含曲面的一个小邻域中. 一种半隐式的Crank-Nicholson 格式被用来做时间推进, 中心差分和三阶加权实质无振荡(WENO) 格式被分别用来离散方程中的扩散项和对流项. 分析证明了它在标准的Courant-Friedrichs-Lewy (CFL) 条件下的稳定性. 数值算例显示了它能取得二阶精度.     

线性对流占优扩散问题的交替方向差分流线扩散法

本文将交替方向法与差分流线扩散法(简称FDSD方法)相结合,对于二维线性对流占优扩散问题构造了一种交替方向差分流线扩散格式,给出了格式的实现过程并就稳定性及误差进行了分析．此格式不但实现了对数值求解二维对流扩散方程降维的目的,并且保持了FDSD方法良好的稳定性及高精度阶的基本性质．最后给出数值算例说明算法的有效性．     

求解对流扩散方程的高阶紧致Padé有限体积格式

<正>自然界许多物理现象都可用对流扩散方程来描述,如质量、能量以及动量守恒问题等.实际应用问题中的对流扩散方程往往比较复杂,难以求出精确解,因此研究其数值求解方法具有十分重要的意义.对流扩散方程的经典解法对于解光滑问题可以得到较好的计算结果,但对于大梯度问题以及边界层等问题,会产生较大误差.紧致格式使用较少的模板可以获得较高的精度,因此高精度紧致方法成了近年来的研究热点[1-4].针对已有高阶紧致格式在分辨率和守恒性方面的问题,本文借鉴文献[4][5]中的思想构造了非定常对     

对流扩散方程的数值计算

本文研究了对流扩散方程的一种并行格式.利用一组saul'yev型非对称格式进行二次构造,分别得到了一类并行GE格式和GEL、GER格式;进一步推广,得到绝对稳定的交替分组显式AGE格式,并用数值例子检验AGE格式的数值计算效果.     

变系数对流-扩散方程的交替分段Crank-Nicolson方法

对Saul'yev型格式中的对流项构造了一种新的离散化逼近形式,进而给出了变系数对流-扩散方程的Crank-Nicolson方法.这个方法是绝对稳定的.数值实验表明该方法并行性好,精度高,宜于直接在并行计算机上使用.     

求解对流扩散方程的一致四阶紧致格式

目前,许多高精度差分格式,由于未成功地构造与其精度匹配的稳定的边界格式,不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式,它的边界点的计算格式和内点的计算格式的截断误差主项保持一致,给出了具体内点和边界格式;并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度;在对流占优情况下,本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小,取得了不错的效果,理论结果得到了数值验证;驱动方腔数值结果显示,本文对N-S方程的离散格式具有很好的可靠性,适合对复杂流体流动的数值模拟和研究.     

基于样条插值求解对流扩散方程

首先,基于样条插值和Padé逼近公式,构造了一种求解一维对流扩散方程的高精度紧致差分格式,其截断误差为o(τ~5+h~4).其次,利用Fourier分析方法证明了格式是无条件稳定的.最后,通过数值算例对文中格式的精度进行了数值测试,进一步验证了格式的准确性和稳定性等.     

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．     

对流扩散方程的混合时间间断时空有限元方法

构造并分析二阶对流扩散方程的混合时间间断时空有限元格式．利用混合有限元方法将二阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散低阶方程．证明数值解的稳定性、存在唯一性和收敛性．最后通过数值结果验证该算法的有效性和可行性．     

MULTIGRID METHOD FOR FLUID DYNAMIC PROBLEMS

This paper covers the dynamics problems. The review and some aspects of main development stages of using Multigrid method for fluid multigrid technics are presented. Some approaches for solving Navier-Stokes equations and convection- diffusion problems are considered.     

Two stable methods with numerical experiments for solving the backward heat equation

This paper presents results of some numerical experiments on the backward heat equation. Two quasi-reversibility techniques, explicit filtering and structural perturbation, to regularize the ill-posed backward heat equation have been used. In each of these techniques, two numerical methods, namely Euler and Crank-Nicolson (CN), have been used to advance the solution in time.Crank-Nicolson method is very counter-intuitive for solving the backward heat equation because the dispersion relation of the scheme for the backward heat equation has a singularity (unbounded growth) for a particular wave whose finite wave number depends on the numerical parameters. In comparison, the Euler method shows only catastrophic growth of relatively much shorter waves. Strikingly we find that use of smart filtering techniques with the CN method can give as good a result, if not better, as with the Euler method which is discussed in the main text. Performance of these regularization methods using these numerical schemes have been exemplified.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

抛物型初边值问题的有限元与边界积分耦合的离散化及其误差分析

1．引言边界元方法是近二十几年来迅速发展起来的一类新的偏微分方程的数值方法．它的独特之处是将空间的维数降低一维,从而倍受工程技术人员的青睐,并在工程技术与计算数学领域得到越来越广泛的重视和研究．对椭圆型问题,边界元方法的理论与应用研究已取得丰硕成果;对发展型问题,近年来在理论方面的研究也已取得重要进展［6－11］．但边界元方法难以处理非均质问题,而有限元对各类问题及各种区域具有较好的适应性,将两者结合起来可充分发挥各自的优点．文山提出了一种抛物方程初边值问题的有限元与边界积分的耦合方法,其主要思想是…     

热方程非线性边界条件问题的有限差分方法

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection-diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection-diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.     

A FINITE DIFFERENCE METHOD FOR THE HEAT EQUATION WITH A NONLINEAR BOUNDARY CONDITION

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

Semi-implicit finite volume level set method for advective motion of interfaces in normal direction

In this paper a semi-implicit finite volume method is proposed to solve the applications with moving interfaces using the approach of level set methods. The level set advection equation with a given speed in normal direction is solved by this method. Moreover, the scheme is used for the numerical solution of eikonal equation to compute the signed distance function and for the linear advection equation to compute the so-called extension speed [1]. In both equations an extrapolation near the interface is used in our method to treat Dirichlet boundary conditions on implicitly given interfaces. No restrictive CFL stability condition is required by the semi-implicit method that is very convenient especially when using the extrapolation approach. In summary, we can apply the method for the numerical solution of level set advection equation with the initial condition given by the signed distance function and with the advection velocity in normal direction given by the extension speed. Several advantages of the proposed approach can be shown for chosen examples and application. The advected numerical level set function approximates well the property of remaining the signed distance function during whole simulation time. Sufficiently accurate numerical results can be obtained even with the time steps violating the CFL stability condition.     

Quadrature-free spline method for two-dimensional Navier-Stokes equation

In this paper, a quadrature-free scheme of spline method for two-dimensional Navier- Stokes equation is derived, which can dramatically improve the efficiency of spline method for fluid problems proposed by Lai and Wenston（2004）. Additionally, the explicit formulation for boundary condition with up to second order derivatives is presented. The numerical simulations on several benchmark problems show that the scheme is very efficient.     

On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation

In this paper, we study the initial-boundary value problem of the usual Rosenau-RLW equation by finite difference method. We design a conservative numerical scheme which preserves the original conservative properties for the equation. The scheme is three-level and linear-implicit. The unique solvability of numerical solutions has been shown. Priori estimate and second order convergence of the finite difference approximate solutions are discussed by discrete energy method. Numerical results demonstrate that the scheme is efficient and accurate.