Newton Linearized Methods for Semilinear Parabolic Equations

In this study, Newton linearized finite element methods are presented for solving semi-linear parabolic equations in two- and three-dimensions. The proposed scheme is a one-step, linearized and second-order method in temporal direction, while the usual linearized second-order schemes require at least two starting values. By using a temporal-spatial error splitting argument, the fully discrete scheme is proved to be convergent without time-step restrictions dependent on the spatial mesh size. Numerical examples are given to demonstrate the efficiency of the methods and to confirm the theoretical results.     

Banach spaces with the Daugavet property

A Banach space is said to have the Daugavet property if every operator of rank satisfies . We show that then every weakly compact operator satisfies this equation as well and that contains a copy of . However, need not contain a copy of . We also study pairs of spaces and operators satisfying , where is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with is as small as possible and give characterisations in terms of a smoothness condition.

     

Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps

It is known that the implied volatility skew of Forex (FX) options demonstrates a stochastic behaviour which is called stochastic skew. In this paper, we create stochastic skew by assuming the spot/instantaneous variance (InV) correlation to be stochastic. Accordingly, we consider a class of Stochastic Local Volatility (SLV) models with stochastic correlation where all drivers – the spot, InV and their correlation – are modelled by processes. We assume all diffusion components to be fully correlated, as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate the capacity of the presented approach in modelling stochastic skew.     

Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium

Convection induced by the selective absorption of radiation is investigated, for the case of an internal heat source that is modelled quadratically with respect to concentration. The growth rate for the linearised system is shown to be real, and a linear instability analysis is performed. To establish conditional and unconditional nonlinear stability results, both the Darcy and Forchheimer models are employed to describe fluid flow. Due to the presence of significant regions of potential subcritical instabilities, the results indicate that linear theory may only be accurate enough to predict the onset of convective motion when the model for the internal heat source is predominantly linear.Received: 6 May 2003, Accepted: 9 August 2003, Published online: 12 December 2003     

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.     

The sufficient and necessary conditions of unconditional stability and the delay bound of the third-order neutral delay differential equation

In this paper,the sufficient and necessary conditions of the unconditional stability.andthe delay bound of the third-order neutral delay differential equation with real constantcoefficients are given.The conditions are brief and practical algebraic criterions.Furthermore,we get the delay bound     

一种结构动力时程分析的积分求微方法

传统采用微分求积(differential quadrature,DQ)法求解动力问题时都是以位移响应作为基本未知量,而将速度响应和加速度响应表示为位移响应的加权和的形式.如此做法需要处理线性方程组或者矩阵方程(Sylvester方程)才能求得动力响应,导出的算法一般为有条件稳定算法.本文利用动力响应的Duhamel积分解,逆用DQ原理,提出了一种计算卷积的高精度显式算法.该算法可以逐时段地求解出动力时程响应,当各时段内DQ节点分布完全一致时,仅须进行一次Vandermonde矩阵求逆计算即可应用于各个时段,一次性获得时段内多个时刻的位移响应值,因而具有计算效率高的优点.通过分析动力方程积分格式,证明本文动力算法传递矩阵的谱半径恒等于1,因而该算法具有无条件稳定特性,且计算过程中不会产生数值耗散. 本文算法的数值精度取决于分析时段内布置的DQ节点数量$N$,具有$N-1$阶代数精度.实际操作时可以取10个甚至更多的DQ节点数,从而获得比较高的数值精度.      

三阶中立型时滞方程无条件稳定的充要条件及时滞界限

本文得到三阶定常中立型时滞方程无条件稳定的充要条件,这些条件是实用的代数判据.此外,还给出了时滞界限.     

Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg-Landau equations

In this paper, a linearized semi-implicit finite difference scheme is proposed to solve the strongly coupled fractional Ginzburg-Landau equations. The difference scheme, which involves three time levels, is unconditionally stable, fourth-order accurate in space, and second-order accurate in time. By using the energy method and mathematical induction, the unique solvability, the unconditional stability, and optimal pointwise error estimate are obtained. Finally, some numerical experiments are presented to validate our theoretical findings.