多辛Dirac方程的高阶整体保能量格式

基于四阶平均向量场方法和拟谱方法构造了Dirac方程的高阶整体保能量格式，利用构造的高阶整体保能量格式数值模拟方程孤立波的演化行为.数值模拟结果表明构造的高阶整体保能量格式可以很好地模拟Dirac方程孤立波的演化行为，并且可以精确地保持方程的整体能量守恒特性.     

解模守恒微分方程的显式平方守恒格式

对具有模守恒的微分方程,经典的显式Runge—Kutta方法和线性多步方法不能保微分方程的模守恒特性．我们利用李群算法和Cayley变换构造了高阶显式平方守恒格式,应用到模守恒的微分方程如Euler方程,Landau—Lifshitz方程,并且与相同阶的显式Runge—Kutta方法在保模守恒和精度方面进行了比较,数值结果表明用李群算法构造的新的显式平方守恒格式能保微分方程模守恒的特性且它和相应Runge—Kutta方法有相同的精度．     

三阶Airy方程满足两个守恒律的非线性差分格式

近年来,学者们对发展型偏微分方程设计了一种能保持多个守恒律的数值方法,这类方法无论在解的精度还是长时间的数值模拟方面都表现出非常好的性质.将这类思想应用到三阶Airy方程,即三阶散射方程,对其设计了满足两个守恒律的非线性差分格式.该格式不仅计算数值解,同时计算数值能量,并且保证数值解和数值能量同时守恒.从数值结果可以看出,该格式在长时间的数值模拟中具有更好的保结构性质.     

非线性薛定谔方程的平均向量场方法

利用平均向量场方法(AVF)对非线性薛定谔方程进行求解, 在理论上得到了一个保非线性薛定谔方程描述的系统能量守恒的AVF格式, 再分别用非线性薛定谔方程的AVF格式和辛格式数值模拟孤立波的演化行为, 并比较两个格式是否保系统能量守恒特性. 数值结果表明, AVF格式也能很好地模拟孤立波的演化行为,并且比辛格式更能保持系统的能量守恒.     

双曲松弛粘性Cahn-Hilliard方程的有限元算法

本文研究双曲松弛粘性Cahn-Hilliard方程的混合有限元数值算法.求解具有双曲松弛项和双阱势能的粘性Cahn-Hilliard方程时,在时间上采用一阶半隐格式进行离散,在空间上采用混合有限元方法进行离散.通过严格的理论分析证明了数值格式的无条件能量稳定性和误差估计,并利用数值实验验证了该方法的有效性.     

非线性振动分析的均向量场法

通过构造向量形式的振动微分方程组，利用均向量场（AVF）法得到振动响应的向量差分迭代格式.该离散格式能够保能量，同时具有二阶精度的特征，从而给出非线性振动问题的均向量场法.介绍了均向量场法的基本步骤.在建立AVF格式时，对于微分方程中若干常见的项，直接给出相应的映射项.应用均向量场法研究了非线性单摆问题和Kepler(开普勒)问题，数值结果说明了该方法保能量和具有长时间求解能力的特性.     

分数阶Cahn-Hilliard方程的高效数值算法

给出了时空分数阶Cahn-Hilliard方程的一个高效数值算法.首先,利用Laplace变换将时空分数阶Cahn-Hilliard方程转化为空间分数阶Cahn-Hilliard方程;然后,结合Fourier谱方法和有限差分法得到一个时间二阶、空间谱精度的高效数值格式;最后,通过数值实验验证本文数值算法的有效性,并验证其满足能量耗散性质和质量守恒定律.     

非线性Cahn-Hilliard方程的拟谱算法

本文对非线性Cahn-Hilliard方程构造了拟谱格式,证明了该格式的收敛性和稳定性,给出了数值例子.     

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.     

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…     

A multigrid method for the Cahn-Hilliard equation with obstacle potential

We present a multigrid finite element method for the deep quench obstacle Cahn-Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ. Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn-Hilliard equation.     

A second order splitting method for the Cahn-Hilliard equation

Summary A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.Research partially supported by NSF Grant DMS-8896141     

Convergence analysis for a second-order elliptic equation by a collocation method using scattered points

A collocation method using scattered points applied to a second-order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials. We show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method.     

A discontinuous Galerkin method for the Cahn-Hilliard equation

The Cahn-Hilliard equation is modeled to describe the dynamics of phase separation in glass and polymer systems. A priori error estimates for the Cahn-Hilliard equation have been studied by the authors. In order to control accuracy of approximate solutions, a posteriori error estimation of the Cahn-Hilliard equation is obtained by discontinuous Galerkin method.     

Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=u^m(1-u)^m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.     

Numerical approximation of the Cahn-Larché equation

Summary Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation - a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. Here, we present a discretization based on finite elements and an implicit Euler scheme. We first show solvability and uniqueness of solutions. Based on an energy decay property we then prove convergence of the scheme. Finally we present numerical experiments showing the impact of elasticity on the morphology of the microstructure.Research supported by DFG Priority Program Analysis, Modeling and Simulation of Multiscale Problems under AR234/5-2 and GA695/2-2     

Finite element scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient

A finite element scheme is considered for the viscous Cahn-Hilliard equation with the nonconstant gradient energy coefficient. The scheme inherits energy decay property and mass conservation as for the classical solution. We obtain the corresponding error estimate using the extended Lax-Richtmyer equivalence theorem.     

On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem

Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L ²-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.     

随机Cahn-Hilliard方程的一种差分逼近

Stochastic Cahn-Hilliard equation is an equation of the field theory for solving the Non-equilibrium dynamics problem in a weak state and is a case of nonlinear Langevin equation.In this paper,using the ackward difference method(BDM),a numerical solution of the stochastic C-H equation is proposed and using the Ito formula,probability and the martingale theory,the convergence of the numerical process is proved in the meaning of mean square.