Hamilton-Jacobi方程的小波Galerkin方法

本文选择Daubechies小波尺度函数空间作为Galerkin方法的测试函数空间,并将其应用于Hamilton-Jacobi方程,得到了求解Hamilton-Jacobi方程的小波Galerkin方法的数值格式．由于小波在时间和频率上的局部性,本方法适用于处理具有奇异解的问题,可以有效地防止数值振荡．数值试验显示,本方法是有效的．     

Kuramoto-Sivashinsky方程的线性化L_∞-差分方法

利用有限差分方法研究Kuramoto-Sivashinsky方程初边值问题的数值解.首先,给出了二阶线性化隐式差分格式,该格式在每一时间层均为线性方程组.其次,给出差分格式的守恒性和数值解的有界性.第三,证明差分格式在最大模意义下的收敛性.最后,通过数值算例验证差分格式的收敛阶,并数值模拟方程的混沌解.     

非结构网格上求解二维H-J方程的一种WENO格式

基于WENO(Weighted Essentially Non-Oscillatory)的思想,提出了一种在非结构网格上求解二维Hamilton-Jacobi(简称H-J)方程的数值方法.该方法利用Abgrall提出的数值通量,在每个三角形单元上构造三次加权插值多项式,得到了一个求解H-J方程的高阶精度格式.数值实验结果表明,该方法计算速度较快,具有较高的精度,而且对导数间断有较高的分辨率.     

一维单个守恒律的一类二阶、几乎熵耗散的格式

本文考虑一维单个守恒律方程，对其设计了一个基于熵耗散的非线性守恒型差分格式．本格式的数值流函数是Lax-Freidrichs格式和Lax-Wendroff格式数值流函数的凸组合，凸组合中的系数是由考虑耗散熵来决定的．这样在解的光滑区域内，格式几乎、甚至完全是Lax-Wendroff格式，而在解的间断处，格式几乎、甚至完全是Lax—Freidrichs格式．从而消除了间断附近的非物理振荡，实现了计算的非线性稳定性．理论分析表明本格式在解的非极值点处是二阶精度的，而在解的极值点处至少有一阶精度．数值试验表明格式是有效的．     

关于广义KPP方程的数值解

广义KPP(Kolmogorov-Petrovskii-Piskunov)方程是一个积分微分方程.为了要研究其数值解,我们首先将该方程转化为一个非线性双曲型方程,然后构造了一个线性化的差分格式,得到了差分格式解的存在唯一性,利用能量不等式证明了差分格式二阶收敛性和关于初值的无条件稳定性,数值结果验证了本文提出的方法.     

三维涡度方程双向周期问题的拟谱─差分解法

本文构造了三维涡度方程双向周期问题的Ｆｏｕｒｉｅｒ拟谱─差分格式，其数值解满足半离散守恒律．文中分析了格式的广义稳定性和收敛性．数值例子表明这类格式的优越性．     

Sine-Gordon方程的多辛Leap-frog格式

非线性发展方程由于具有多种形式的解析解而吸引着众多的研究者,借助多辛保结构理论研究了Sine-Gordon方程的多辛算法．利用Hamilton变分原理,构造出了sine-Gordon方程的多辛格式;采用显辛离散方法得到了Leap-frog多辛离散格式,该格式满足多辛守恒律;数值结果表明leap-frog多辛离散格式能够精确地模拟sine-Gordon方程的孤子解和周期解,模拟结果证实了该离散格式具有良好的数值稳定性．     

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.     

广义混合非线性Schrdinger方程的拟谱方法

本文讨论了广义混合非线性Schrdinger 方程的周期初值问题,构造了守恒的半离散Fourier 拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

广义混合非线性Schrödinger方程的拟谱方法

本文讨论了广义混合非线性Schrodinger方程的周期初值问题,构造了守恒的半离散Fourier拟谱格式,对其近似解进行了先验估计,并证明了格式的收敛性.证明了该方程存在孤立子解,并给出其孤立子解的精确表达式.研究了线性化方程的稳定性问题,即在初值有扰动的情况下,该方程只有振荡解和鞍点.最后,通过数值例子验证了格式的可信性,数值计算表明,本格式时间方向可取大步长且是长时间稳定的,我们还计算了孤立子解,并绘出了在初值有扰动的情况下,相空间的轨线图.     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

Finite volume schemes for Hamilton–Jacobi equations

Summary. We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity solution of the Hamilton–Jacobi equation. Received February 25, 1998 / Published online: June 29, 1999     

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Summary. Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given. Received August 1, 1993 / Revised version received November 7, 1994     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Numerical schemes for Hamilton-Jacobi equations on unstructured meshes

Summary.  In this paper, a numerical scheme is presented by applying the finite element method to the viscosity equations of the Hamilton-Jacobi equations on unstructured meshes. By improving the finite element scheme, another numerical scheme is constructed. Under certain limitations, the numerical solutions of the two schemes converge to the viscosity solutions of the Hamilton-Jacobi equations. The latter numerical scheme has weaker restrictions than the former scheme for convergence. Numerical examples are provided to test the stability, convergence and sensitivity to different meshes. Received November 5, 2001 / Revised version received March 5, 2002 / Published online October 29, 2002 RID="*" ID="*" Current address: Department of Applied Mathematics, University of Petroleum, Dongying 257062, Shandong, P.R.China; e-mail: xianggui_li@sina.com Mathematics Subject Classification (1991): 65M60     

Error estimates for approximation schemes of effective Hamiltonians arising in stochastic homogenization of Hamilton-Jacobi equations

We study approximation schemes for effective Hamiltonians arising in the homogenization of first order Hamilton-Jacobi equations in stationary ergodic settings. In particular, we prove error estimates concerning the rate of convergence of the approximated solution to the effective Hamiltonian. Our main motivations are front propagation problems, but our results can be generalized to other types of Hamiltonians.     

Hamilton-Jacobi方程的单调有限元格式

A monotone finite element scheme is obtained by applying the finite element method to the viscosity equation of the Hamilton-Jacobi equation on unstructured meshes. Under some constraints, we show that this scheme is monotone and its numerical solution converges to the viscosity solution of the Hamilton-Jacobi equa-tion. Numerical examples test the stability and the convergence of this scheme.     

On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems

The paper is devoted to the development of the viscosity approach to the generalized solution of functional Hamilton-Jacobi type equations with coinvariant derivatives and a nonanticipatory Hamiltonian. These equations are naturally connected to problems of dynamical optimization of hereditary systems and, as compared with classical Hamilton-Jacobi equations, possess a number of additional peculiarities stipulated by the aftereffect. The definition of a viscosity solution that takes the above peculiarities into account is given. The consistency of this definition with the notion of a classical solution and with the minimax approach to the generalized solution is substantiated. The existence and uniqueness theorems are proved.     

Construction of Canonical Difference Schemes for Hamiltonian Formalism via Generating Functions

This paper discusses the relationship between canonical maps and generating functions and gives the general Hamilton-Jacobi theory for time-independent Hamiltonian systems. Based on this theory, the general method — the generating function method — of the construction of difference schemes for Hamiltonian systems is considered. The transition of such difference schemes from one time-step to the next is canonical. So they are called the canonical difference schemes. The well known Euler centered scheme is a canonical difference scheme. Its higher order canonical generalisations and other families of canonical difference schemes are given. The construction method proposed in the paper is also applicable to time-dependent Hamiltonian systems.     

用基于水平集方法的自适应运动网格变形方法解Hamilton-Jacobi方程组

A numerical scheme is presented for solving the Hamilton-Jacobi equation by applying adaptive moving grid methods of level-set-based deformation methods. Two numerical examples are given, which demonstrate the accuracy and efficiency of computing“extreme”and“spikes”of solutions to the Hamilton-Jacobi equation.