【作者】 赵然；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 计算数学， 2006， 硕士

【摘要】 奇异摄动问题广泛地存在于化学动力学、自动控制、电子系统等一系列高科技领域的数学模型中,这类问题往往具有重要性和特殊的计算复杂性,发展和深化针对各种数值方法的定量收敛理论受到了各国科学家的广泛关注。奇异摄动初值问题是一类特殊的刚性问题,其特殊的结构特点使得这类问题不能被B-理论满意地覆盖。这一问题主要包括单刚性(刚性仅由小参数引起)和多刚性(刚性由小参数和其它因素引起)奇异摄动初值问题两大类。目前,对于最简单形式的单参数奇异摄动初值问题的数值方法国内外已有一些研究,Hairer,Lubich,Strehmel,Schneider,肖爱国,甘四清和孙耿等人基于直接法或ε-渐进展开法两种不同的方法分别给出了许多关于线性多步方法,Runge-Kutta方法,Rosenbrock型方法,一般线性方法等关于奇异摄动经典初值问题的收敛性结果。然而对于同时具备离散型延迟和分布型延迟的一类奇异摄动积分微分系统初值问题数值方法收敛性的研究,国内外尚未见到这方面的结果。因此,对该领域的研究无疑具有重要的理论意义和广阔的应用前景。本文首先探讨了一类延迟奇异摄动积分微分系统的指数稳定性质,第三章和第四章分别致力于讨论线性多步方法关于单刚性和多刚性两类问题的定量收敛性,给出了其整体误差分析。每章结尾的数值试验结果证明了本文理论结果的正确性。更多还原

【Abstract】 Singular perturbation problems widely arise in many practical applications, such aschemical kinetics, automatic control, and electrical circuits et.al. By view of the specialcomplexity and importance of this problems, many scientists from all over the worldhave paid extensive attention to develop and deepen quantitative convergence theory ofnumerical methods for these important typical subclasses of stiff problems.Initial value problems of ordinary differential equations in singular perturbation formare of the class of stiff problems which can’t be satisfactorily covered by B-theory becauseof their very special structure. They mainly include two subclasses: singly stiff singularperturbation problems (their stiffness is only caused by some small parameters) and multi-ply stiff singular perturbation problems (their stiffness is caused by some small parametersand some other factors). At present, there exist many researches into numerical methodsfor typical singular perturbation problems. Many important and interesting results on theconvergence of linear multistep methods, RungeffKutta methods, Rosenbrock methodsand general linear methods applied to the classical singular perturbation problems havebeen given by many authors by means of two different approaches ff direct approach and-expansion approach.But up to now, there exits no results of numerical methods for singular perturbationintegral differential problems with delays which often arise in many practical applications,such as chemical kinetics, automatic control, electrical circuits and nuclear reactor kineticset.al. Therefore, a deep research into this field is of important theoretical meanings andvast applied prospects.In this paper, we first proved the stability property of a class of singular perturba- tion delay volterra integral differential systems(SPDVIDs). Then, we have gained somequantitative convergence results for linear multistep methods applied to both singly stiffsingular perturbation problems and multiply stiff singular perturbation problems in chap-ter 3 and 4. In the end of each chapter, we also gave some numerical test which confirmedour theoretical results.更多还原