Uniform Convergence Analysis of Finite Difference Scheme for Singularly Perturbed Delay Differential Equation on an Adaptively Generated Grid

<正>Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.     

A COMPACT UPWIND SECOND ORDER SCHEME FOR THE EIKONAL EQUATION

We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.     

二阶椭圆问题新的混合元格式

本文基于二阶椭圆问题一种新的混合变分形式,给出同时满足强椭圆性和B-B条件的任意次的求解格式.理论分析表明这些单元论证简单而且用了较少的自由度达到最优误差估计.同时我们还给出了它们在各向异性网格下的误差估计.     

投影三角分解法定价带随机波动率的美式期权

考虑数值求解Heston随机波动率美式期权定价问题,通过在空间方向采用中心差分格式离散二维偏微分算子,在时间方向利用隐式交替方向格式,将美式期权定价问题转化成求解每个时间层上的若干个线性互补问题.针对一般美式期权定价模型离散得到的线性互补问题,构造出投影三角分解法进行求解,并在理论上给出算法的收敛条件.数值实验表明,所构造的数值方法对于求解美式期权定价问题是有效的,并且优于经典的投影超松弛迭代法和算子分裂方法.     

征稿简则

简介《数学进展》创刊于1955年,首任主编是华罗庚教授。本刊是由中国数学会主办(北京大学数学科学学院承办)的一个综合性数学刊物,主要刊登纯粹数学和应用数学方面的综述文章和创造性学术论文。其宗旨是介绍数学各分支的发展动态,反映数学研究的最新成果,促进国内外的学术交流,推动我国数学研究的发展。主要内容1.介绍数学中的热点研究方向或重大问题的研究现状的综述文章。这类文章的     

一类非线性Schr(o)dinger方程的非协调有限元分析

在半离散格式下研究一类带幂次非线性项的Schr(o)dinger方程的非协调矩形EQrot1元方法.直接利用插值技巧和该单元的两个特殊性质(相容误差比插值误差高一阶及其插值算子与传统的Ritz投影是一致的),给出相应的收敛性分析及误差估计.     

各向异性网格下抛物方程一个新的非协调混合元收敛性分析

本文将 Crouzeix-Raviart 型非协调线性三角形元应用到抛物方程,建立了一个新的混合元格式.在抛弃传统有限元分析的必要工具 Ritz 投影算子的前提下,直接利用单元的插值性质和导数转移技巧, 分别得到了各向异性剖分下关于原始变量u 的H^-1-模和积分意义下L²-模以及通量p=-▽u 在L²-模下的最优阶误差估计.数值结果与我们的理论分析是相吻合的.     

大变形网格上时间二阶精度扩散方程的保正格式

本文基于已有的连续扩散通量的两点非线性离散格式,构造了2D非稳态扩散方程大变形网格上的两层非线性有限体积格式.该格式利用Crank-Nicolson (C-N)方法的思想在时间方向获得了二阶精度.由于所得代数方程组的系数矩阵的转置是M矩阵,从而能够保持解的正性,并利用Brouwer不动点定理证明了格式解的存在性.数值实验结果表明,在较大时间步长下,该格式具有二阶计算精度.     

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