ON THE CONVERGENCE OF GODUNOV SCHEME FOR NONLINEAR HYPERBOLIC SYSTEMS

51. IntroductionWe consider the one dimensional Cauchy problem for an n x n system of the formHere A(u) is a smooth matrix valued map from a domain U C R" into R", and (x, t) eR x R . The system is assumed to be strictly hyperbolic, i.e. the matrix A(u) has n realand strictly different eigenvalues at each point u C U.We note that even for smooth data a classical solution is only defined locally in time.In general the solution will develop discontinuities in finite time and it is not cle…     

非线性抛物型方程组离散向量解的收敛性

应用离散泛函分析方法证明了非线性抛物型方程组的隐格式离散向量解的收敛性，同时得到微分方程组弱解的存在性．     

NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION

We consider numerical methods to solve the Allen-Cahn equation using the second-order Crank-Nicolson scheme in time and the second-order central difference approach in space.The existence of the finite difference solution is proved with the help of Browder fixed point theorem.The difference scheme is showed to be unconditionally convergent in L∞ norm by constructing an auxiliary Lipschitz continuous function.Based on this result,it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size.The numerical experiments also verify the reliability of the method.     

THE CAUCHY PROBLEM FOR A CLASS OF DOUBLY DEGENERATE NONLINEAR PARABOLIC EQUATION

This article studies the Cauchy problem for a class of doubly nonlinear degenerate parabolic equations . Under certain conditions, the author considers its regularized problem and establishes some estimates. On the basis of the estimates, the existence and uniqueness of the generalized solutions in BV space are proved.     

非线性抛物型方程初边值问题解的Blow up性质

本文研究一类非线性抛物型方程具有非线性边界条件的初边值问题.利用抛物型方程最大值原理和凸性方法证明了该问题的解在有限时间内爆破.推广了文献[7-9]的结果.     

CONVERGENCE RATES FOR DIFFERENCE SCHEMES FOR POLYHEDRAL NONLINEAR PARABOLIC EQUATIONS

We build finite difference schemes for a class of fully nonlinear parabolic equations. The schemes are polyhedral and grid aligned. While this is a restrictive class of schemes, a wide class of equations are well approximated by equations from this class. For regular （C2,α） solutions of uniformly parabolic equations, we also establish of convergence rate of O（α）. A case study along with supporting numerical results is included.     

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.     

广义Newton法求解非线性互补问题

１引言考虑非线性互补问题ＮＣＰ（ｆ）：的求解,即我们要寻求某ｘ∈Ｒｎ,使其满足（１．１）．其中映射ｆ：Ｒｎ→Ｒｎ为具有连续Ｆ－导数的非线性映射．众所周知,问题（１．ｌ）可以等价地转化为Ｂ－可微方程组：求解,其中：容易证明,由（１．３）定义的映射Ｇ处处Ｂ－可微,且其在点ｘ∈Ｒｎ处的Ｂ－导数ＢＧ（ｘ）为而对于问题（１．２）（１．３）,我们希望直接用经典的广义Ｎｅｗｔｏｎ法进行求解．但是,由于由（１．３）定义映射Ｇ在（１．１）的解ｘ∈Ｒｎ处,没有可逆的强Ｆ－导数存在,因此,关于算法（１．５）（１．６）…     

椭圆型奇异摄动问题差分格式的一致收敛性分析

本文研究了一类椭圆型奇异摄动问题.利用Bakhvalov-Shishkin网格上的差分方法,获得了数值解一致一阶收敛于真解的结果.     

二维非线性抛物型微分方程的一个三层差分格式

1.引言 对于非线性抛物型微分方程第一边值问题: ?其中Ω:{0≤x,y≤1},0相似文献   

NEW ALTERNATING DIRECTION FINITE ELEMENT SCHEME FOR NONLINEAR PARABOLIC EQUATION

A new alternating direction (AD) finite element (FE) scheme for 3-dimensional nonlinear parabolic equation and parabolic integro-differential equation is studied. By using AD, the 3-dimensional problem is reduced to a family of single space variable problems, calculation work is simplified; by using FE, high accuracy is kept; by using various techniques for priori estimate for differential equations such as inductive hypothesis reasoning, the difficulty arising from the nonlinearity is treated. For both FE and ADFE schemes, the convergence properties are rigorously demonstrated, the optimal H1-and L2-norm space estimates and the 0((△t)2) estimate for time variable are obtained.     

A SINGLE STEP SCHEME WITH HIGH ACCURACY FOR PARABOLIC PROBLEM

1 IntroductionConsider one-dimellsional parabolic problem in fl = (0, 1)and its weak formulatioll: find u(t) E S0 = {v 6 H'(fl), v(0) = 0} 8uch thatwhere the coefficients a(x) and b(x) are independent of t, Au = --(au')' ha, a(x) 2 ao > 0, b 20, and bilinear fOrm A(u, v) = fol(au'V' buv)dx i8 So-coercive, i.e. A(u, v) 2 ullu1li, u ESO, u > 0.Make in n = (0, 1) a subdivision: xo = 0 < x1 < x2 <.' < xn = 1. Set an elenlentry = (xj--l, xj), nddpoint xj--1/2 = (ry n--1)/2 and steplengt…     

关于广义Newton法的收敛性问题

本文在较弱的条件下,证明了Ｂ－可微方程组的广义Ｎｅｗｔｏｎ法的局部超线性收敛性,为该算法直接应用于非线性规划问题、变分不等问题以及非线性互补问题等提供了理论依据。最后,本文给出了广义Ｎｅｗｔｏｎ法付之实践的具体策略。数值结果表明,算法是行之有效的。     

求解互补问题的Newton方法的半局部收敛性

A semilocal convergence theorem is given for Newton method solving complementarity problems, which is identical in form to the standard Kantorovich theorem. All the convergence condition can be verified computationally.     

THE CONVERGENCE OF A DOMAIN DECOMPOSITION ALGORITHM FOR PARABOLIC PROBLEMS

At recent, Hourgat et gave a domain decomposition algorithm for elliptic problems which can be implemented in parallel. Many numerical experiments have illustrated its efficiency. In the present paper, we apply this algorithm to solve the discrete parabolic problems, analyse its convergence and show that its convergence rale is about (1 - 2p + σp2 ) which is nearly optimal and independent of the parameter τ, where σ τ O((1 +H )(1 + ln(H / h))2 ). 0 < p < 1 / σ,τ,h,H are the time step size, finite element parameter and subdomain diameter, respectively.     

Захаров方程周期边界条件一类有限差分格式的收敛性和稳定性

§1.前言 自从[1]中提出axapoB方程后,由于它具有孤立子解,且有不同于KdV方程孤立子的一些性质,同时也可考察孤立子和其他波(例如声波)的相互作用,对激光打靶出现密度坑的物理现象提出了比较合理的解释,因而,引起了人们很大的兴趣。它的数值求解,已在[2-4]等中用有限差分法进行了数值计算,且有不少计算结果。本文主要从计算理论上证明一类和它相应的差分格式的收敛性和稳定性。     

BV SOLUTIONS OF DIRICHLET PROBLEM FOR A CLASS OF DOUBLY NONLINEAR DEGENERATE PARABOLIC EQUATIONS

The uniqueness and existence of BV solutions to Dirichlet problem of doubly degenerate parabolic equations of the following form au/at=div(A(|↓△B(u)|↓△B(u))in QT=Ω&#215;(0,T) are studied     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

MIXED FINITE ELEMENT METHODS FOR A STRONGLY NONLINEAR PARABOLIC PROBLEM

1.IntroductionFOrsecondorderellipticproblems,themixedmethodwasdescribedandanalyzedbymanyauthors[1--3]inthecaseoflinearequationsindivergenceform,aswellasin[4,5]forquasilinearornonlinearproblemsindivergenceform.JohnsonandThorn6e[6]consideredalternativeproofSofthepreviouslyknownerrorestimatesforsuchmethodsintheellipticcase.Theyalsoanalyzedthemixedfiniteelementmethodforthepaxabolicequationgivenbypt--ho~f.Garcia[7]studiedtheconvergenceofmixedfiniteelementapproximationstoquasilinearparabolicequati…