三维热传导型半导体问题的交替方向特征有限元方法及理论分析

１．引言 三维热传导型半导体器件瞬态问题的数学模型由四个非线性偏微分方程描述 ［１,２］．工程研究中一般考虑绝流边条件,由于绝流条件可以看作一反射条件来处理、为了数值分析方便,我们在此考虑三维周期问题： 其中,　＝［０,１］３,未知函数是电子位势　;电子,空穴浓度ｅ,ｐ;温度函数Ｔ．方程（１,１）－（１.４）中出现的系数均有正的上下界,且是　周期的． ａ＝Ｑ／ε,Ｑ,ε分别表示电子负荷和介电系数,均为正常数．Ｎ（ｘ）是给定的函数．Ｄｓ（ｘ）为扩散系数,μｓ（ｘ）为迁移率,ｓ＝ｅ,Ｐ．Ｒ（ｅ,ｐ,Ｔ）…     

积分微分方程线性多步方法的散逸性

研究一类积分微分方程线性多步方法（p,σ）的散逸性．当积分项用复合求积公式逼近时,证明了线性多步方法是有限维散逸的．这说明该方法很好地继承了系统本身所具有的重要性质．这一结论为数值求解这一类微分方程提供了更多的选择．     

三维热传导型半导体问题的特征混合元方法和分析

本文研究三维热传导型半导体态问题的特征混合元方法及其理论分析,其数学模型是一类非线性偏微分方程的初边值问题,对电子位势方程提出混合元逼近,对电子,空穴浓度方程笔挺表限元逼近;对热传导方程采用对时间向后差分的Ｇａｌｅｒｋｉｎ逼近,应用微分方程先验估计理论和技巧得到了最优阶Ｌ＾２误差估计。     

Multistep Hermite collocation methods for solving Volterra Integral Equations

In this paper, we propose a new class of multistep collocation methods for solving nonlinear Volterra Integral Equations, based on Hermite interpolation. These methods furnish an approximation of the solution in each subinterval by using approximated values of the solution, as well as its first derivative, in the r previous steps and m collocation points. Convergence order of the new methods is determined and their linear stability is analyzed. Some numerical examples show efficiency of the methods.     

非矩形区域上半导体问题的交替方向有限元方法

Transient behavior of semiconductor with heat-conduction on nonrectangular is studied using isoparametric elements and an approximation to the Jacobian of the isoparametric map.Concentration and heat-conduction equations are solved by alternating-direction methods and electric potential equation is approximated by finite element method.Optimal order error estimates in L^2 are demonstrated using the theory and technique of a prior estimate of differential equation.     

Über die Koerzitivität gewöhnlicher Differenzenoperatoren und die Konvergenz von Mehrschrittverfahren

Summary For linear ordinary difference operators under general boundary conditions a so called coerciveness inequality is proved. As an application the solutions and their difference quotients up to the order of the approximated differential equation of stable and consistent multistep difference methods are shown to converge.

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Herrn Prof. Dr. Dr. h.c. L. Collatz zum 60. Geburtstag gewidmet     

Sixth-Order Compact Extended Trapezoidal Rules for 2D Schrödinger Equation

Based on high-order linear multistep methods (LMMs), we use the class of extended trapezoidal rules (ETRs) to solve boundary value problems of ordinary differential equations (ODEs), whose numerical solutions can be approximated by boundary value methods (BVMs). Then we combine this technique with fourth-order Padé compact approximation to discrete 2D Schrödinger equation. We propose a scheme with sixth-order accuracy in time and fourth-order accuracy in space. It is unconditionally stable due to the favourable property of BVMs and ETRs. Furthermore, with Richardson extrapolation, we can increase the scheme to order 6 accuracy both in time and space. Numerical results are presented to illustrate the accuracy of our scheme.     

MULTISTEP DISCRETIZATION OF INDEX 3 DAES

1. IntroductionIn this paper, we will consider the multistep discrezations of the differential--algebraicequations (DAEs) in Hessenberg formwhere F e AN M M R", K E AN M L - AM, G E AN - RL, the initial value(yo, ic, no) at xo are assumed to be consistent, i.e., they satisfyWe supposes, F, G and K are sufficiently differentiable, and thatin a neighbourhood of the solution. Such problems often appear in the simulation ofmechanical systems with constraints and the singularly perturbed…     

A New Class of Theorems of the Alternative

An estimation problem for a random set that is a reachable domain of the Ito differential equation with respect to its initial data is considered. The Markov property of the reachable set in the space of closed sets is proved. For the purposes of numerical solution, a random initial set of the differential equation is approximated by a finite set on an integer multidimensional grid, and the differential equation is replaced by a multistep Markov chain. Examples are considered.     

On a multistep method approximating a linear sectorial evolution equation

This paper concerns the linear multistep approximation of anabstract dissipative linear sectorial evolution equation ona Banach space X. We study how well the semigroup generatedby a sectorial operator A is approximated by the numerical semigroupgenerated by a q-step, strictly A ()-stable multistep method.An optimal order error bound is obtained.     

CHARACTERIZATIONS OF SYMMETRIC MULTISTEP RUNGE-KUTTA METHODS

Some characterizations for symmetric multistep Runge-Kutta(RK) methods are obtained. Symmetric two-step RK methods with one and two-stages are presented. Numerical examples show that symmetry of multistep RK methods alone is not sufficient for long time integration for reversible Hamiltonian systems. This is an important difference between one-step and multistep symmetric RK methods.     

Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.     

Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.     

On the solution of heat-conduction problems for thermosensitive bodies

We illustrate methods of constructing analytic-numerical solutions of nonsteady heat-conduction problems for thermosensitive bodies under convective heat transfer, and also two-dimensional steady-state heat-conduction problems for piecewise-homogeneous bodies. Translated fromMatematichni Metodi i Fiziko-Mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 36–44.     

求解非线性刚性初值问题的隐显线性多步方法的误差分析

隐显线性多步方法由隐式线性多步方法和显式线性多步法组合而成.本文主要讨论求解满足单边Lipschitz条件的非线性刚性初值问题和一类奇异摄动初值问题的隐显线性多步方法的误差分析.最后,由数值例子验证了所获的理论结果的正确性及方法处理这两类问题的有效性.     

半导体设备热传输中的隐式-显式多步有限元法

考虑热引导半导体设备中的传输行为，用一个有限元法离散电子位势所满足的Rpoisson方程；用隐式－显式多步有限元法处理电子密度和空洞密度满足的两个对流－扩散方程，热传导方程用隐式多步有限元法离散，推导了优化的L^2范误差估计。     

Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the First Kind

Fractional powers of linear multistep methods are suggestedfor the numerical solution of weakly singular Volterra integralequations of the first kind. The proposed methods are convergentof the order of the underlying multistep method. The stabilityproperties are directly related to those of the multistep method.     

热传导型半导体问题的配置方法及误差分析

热传导型半导体瞬态问题的数学模型是一类非线性偏微分方程的初边值问题．电子位势方程是椭圆型的，电子、空穴浓度方程及热传导方程是抛物型的．该文给出求解的配置方法，得到次优犔２模误差估计，并将配置法和Ｇａｌｅｒｋｉｎ有限元方法进行数值结果比较．     

半导体问题的向后差分配置法及误差分析

Collocation method is put forward to solve the semiconductor problem with heat-conduction, whose mathematical model is described by an initial and boundary problem for a nonlinear partial differential equation system. One elliptic equation is for the electric potential, and three parabolic equations are for the electron concentration, hole concentration and heat-conduction. Using the prior estimate and technique of differential equations, we obtained almost optimal error estimates in L2.     

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.