守恒型扩散方程非线性离散格式的性质分析和快速求解

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.     

修正相场晶体(MPFC)模型的二阶无条件能量稳定数值格式

本文研究修正的相场晶体(MPFC)方程的二阶无条件能量稳定数值格式.首先,利用标量辅助变量(SAV)法和二阶向后欧拉(BDF2)公式,得到了一个数值格式;其次,给出了能量耗散定律,严格证明了该数值格式是质量守恒的,唯一可解的,无条件能量稳定的;最后,通过数值实例验证了格式的精度和稳定性.     

Cahn-Hilliard方程的新型二阶稳定化半隐有限元方法

基于Crank-Nicolson/Adams-Bashforth离散,一种新型的二阶稳定化半隐有限元格式被建立用来求解Cahn-Hilliard方程.在此格式中,通过添加一个新型的二阶稳定项,得到一个满足离散的能量耗散定律的线性系统.空间离散考虑Galerkin有限元方法,从而获得时空全离散格式.算法的稳定性被考虑,同时给出相应的误差估计.理论结果表明,所提出的算法具有二阶精度.最后,数值算例验证所提算法的有效性.     

抛物型方程基于POD方法的时间二阶精度CN有限元降维格式

将特征正交分解(proper orthogonal decomposition, 简记为POD) 方法应用于抛物型方程通常的时间二阶精度Crank-Nicolson (简记为CN) 有限元格式, 简化其为一个自由度极少的时间二阶精度CN 有限元降维格式, 并给出简化的时间二阶精度CN 有限元解的误差分析. 数值例子表明在简化的时间二阶精度CN 有限元解和通常的时间二阶精度CN 有限元解之间的误差足够小的情况下, 简化的时间二阶精度CN 有限元格式能大大地节省自由度, 而且时间步长可以比时间一阶精度的格式取大10 倍, 以至能更快计算到所要时刻数值解, 减少计算机计算过程的截断误差, 提高计算速度和计算精度,从而验证降维时间二阶精度CN 有限元格式用于解类似于抛物型方程的时间依赖方程是很有效的.     

抛物型方程基于POD方法的时间二阶中心差的时间二阶精度简化有限元格式

本文将特征正交分解(Proper Orthogonal Decomposition,简记为POD)方法应用于抛物型方程通常时间二阶中心差的时间二阶精度有限元格式(简称为通常格式),简化其为一个自由度极少但具有时间二阶精度的有限元格式,并给出简化的时间二阶中心差的时间二阶精度有限元格式(简称为简化格式)解的误差分析.数值...     

Cahn-Hilliard方程非条件能量稳定二阶方法的稳定性和收敛性分析

在这项工作中,我们构建了一种非条件能量稳定的高效不变能量四分法(IEQ)来求解Cahn-Hilliard方程.所构建的数值格式是线性的、具有二阶时间精度和非条件能量稳定性.我们仔细分析了数值格式的唯一可解性、稳定性和误差估计.结果表明,所构建的格式满足唯一可解性、非条件能量稳定性和时间方向的二阶收敛性.通过大量的二维和三维数值实验,我们进一步验证了所提出格式的收敛阶、非条件能量稳定性和有效性.     

Sobolev方程的CN全离散化有限元格式

首先给出Sobolev方程关于时间二阶精度的Crank-Nicolson(CN)时间半离散格式,然后直接从时间二阶精度的CN时间半离散格式出发,构造CN全离散化的有限元格式,并给出这种时间二阶精度的CN全离散化有限元解的误差估计.本文研究方法使得理论证明变得更简便, 也是处理Sobolev方程的一种新的尝试.     

线性双曲组混合问题的差分法

针对一组线性常系数双曲方程组的初边值问题讨论了一种有效差分格式。该格式为二阶精度，绝对稳定，虽为隐式，但计算量仅有最简单的显式计算量，格式结构简单，使用方便。     

一维定常型对流占优扩散方程的一类迎风有限体积格式

本文针对一维定常型对流占优扩散方程提出了一类迎风有限体积格式 .该格式对对流项具有二阶精度 ,对扩散项保持一阶精度 ,符合对流占优扩散问题强对流、弱扩散的特点 .     

广义Rosenau-KdV方程的高精度守恒差分格式

对广义Rosenau-KdV方程提出一种在时间层和空间层上分别具有二阶和四阶精度的三层线性差分格式,所建格式是离散质量守恒和离散能量守恒的,利用离散能量法证明了差分格式的可解性、收敛性和稳定性.数值实验验证了该格式的精度和守恒性.     

一类多维非线性反应扩散方程有限差分格式的稳定性问题(英文)

本文研究了一类多维线性反应扩散方程差分格式的稳定性.利用量未知元方法,建立了具有增量未知元的有限差分格式;然后利用非线性Galerkin方法,得到该差分格式的稳定性条件.通过对该格式的稳定性分析,说明和经典的差分格式的稳定性相比较,带有增量未知元的有限差分格式的稳定性得到了提高.     

二维半线性反应扩散方程的交替方向隐格式

本文研究一类二维半线性反应扩散方程的差分方法.构造了一个二层线性化交替方向隐格式.利用离散能量估计方法证明了差分格式解的存在唯一性、差分格式在离散H~1模下的二阶收敛性和稳定性.最后给出两个数值例子验证了理论分析结果.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

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.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

Finite difference approximate solutions for the Cahn‐Hilliard equation

In this article, we analyze a Crank‐Nicolson‐type finite difference scheme for the nonlinear evolutionary Cahn‐Hilliard equation. We prove existence, uniqueness and convergence of the difference solution. An iterative algorithm for the difference scheme is given and its convergence is proved. A linearized difference scheme is presented, which is also second‐order convergent. Finally a new difference method possess a Lyapunov function is presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 437–455, 2007     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

New conservative difference schemes for a coupled nonlinear Schrödinger system

In this paper, two conservative difference schemes for solving a coupled nonlinear Schrödinger (CNLS) system are numerically analyzed. Firstly, a nonlinear implicit two-level finite difference scheme for CNLS system is studied, then a linear three-level difference scheme for CNLS system is presented. An induction argument and the discrete energy method are used to prove the second-order convergence and unconditional stability of the linear scheme. Numerical examples show the efficiency of the new scheme.     

二维热传导方程的三层显式差分格式

对二维热传导方程构造了一个稳定的三层显式差分格式求其数值解,其背景源于高维热力学反问题迭代算法中对正问题小计算量算法的需求。首先建立一个含参数的一般差分格式去逼近微分方程,并得到了最优截断误差。然后导出了参数应满足的条件以保证差分格式的稳定性。最后给出了数值的例子并和其它算法进行比较,说明了格式在精度上的有效性和计算量上的优越性。     

扩散系数为矩阵形式的两相混溶驱动问题的修正迎风差分格式

1　引　　言油藏数值模拟对油田开发意义重大 .两相不可压缩混溶驱动问题 ,其数学模型是一组非线性偏微分方程 ,其中的压力方程是一椭圆型方程 ,饱和度方程是一对流扩散方程 .由于对流为主的扩散方程具有双曲特性 ,中心差分格式虽关于空间步长具有二阶精度 ,但会产生数值弥散和非物理力学特性的数值振荡 ,使数值模拟失真 .特征方法与标准的有限差分方法结合起来可以较好地反映出对流扩散方程的一阶双曲特性 ,从而减少误差 ,提高计算精度[1 ] .在周期性假定下 ,美国数学家 Jim Douglas,Jr教授分别对压力方程采用混合元格式[2 ] 和五点差分…