几类Volterra泛函微方程数值方法的稳定性分析
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摘要
泛函微分方程(FDEs)在自动控制、生物学、医学、化学、人口学、经济学等众多领域有着广泛应用，其理论和算法研究具有无可置疑的重要性，近三十年来，Volterra泛函微分方程(VFDEs)，特别是其重要子类——延迟微分方程(DDEs)的算法理论研究得到了众多学者的高度关注，取得了大量研究成果．例如在DDEs数值方法线性稳定性研究领域，Barwell、Watanabe、Zennaro、Spijker、in’t Hout、Bellen、Jackiewicz、刘明珠、匡蛟勋、田红炯、张诚坚及胡广大等人作了大量工作，其中主要成果可参见Bellen和Zennaro及匡蛟勋的专著；DDEs数值方法非线性稳定性研究始于1989年Torelli及1992年Bellen和Zennaro的工作，1999年，黄乘明、李寿佛等人在BIT发表的论文指出Torelli稳定性是一个仅有极少数低阶方法才能满足的过于苛刻的概念，并提出了一个新的更为合理的稳定性概念，在此基础上，使得DDEs数值方法非线性稳定性研究得以蓬勃发展。尽管当时的研究仍局限于常延迟、等步长、线性插值及负的Lipschitz常数，但研究对象几乎遍及包括线性多步法和Runge-Kutta方法在内的一切常用算法，获得了大量新的数值稳定性结果。VFDEs数值方法的基于经典Lipschitz条件的收敛性研究已获得大量成果，例如可参见李寿佛1997年的专著，关于DDEs数值方法的经典收敛性研究可参见Oberle、Pesch、Bellen、Zennaro、Tavernini、Arndt、Enright、Feldstein、Neves、Karoui、Vaillancourt、Baker和Paul等人的工作。然而这些收敛性理论仅适用于非刚性问题，不适用于刚性问题。非线性刚性DDEs数值方法的收敛性研究始于张诚坚等人1997的工作，他们提出了Runge-Kutta方法的D-收敛概念，并证明了若干隐式Runge-Kutta方法能满足这一要求。其后，黄乘明等人从他们提出的新的更为合理的数值稳定性概念出发，获得了关于隐式Runge-Kutta方法和一般线性方法的大量D-收敛性结果，近年来，李寿佛在《中国科学》等刊物上发表一系列论文，进一步建立了一般的非线性刚性VFDEs的稳定性理论及其数值方法(包括Runge-Kutta方法和一般线性方法)的B-稳定性与B-收敛性理论，后者统称为数值方法的B-理论，可视为Dahlquist、Butcher、Frank及李寿佛等人所建立的刚性常微分方程数值方法的B-理论的进一步推广，应当指出，这里所建立的新理论比文献中已有的理
Functional differential equations(FDEs) arise widely in the fields of control theory, biology, medicine, chemistry, economics and so on. It is meaningful to investigate the theory and application of numerical methods for FDEs. In recent 30 years, the theory of computational methods for Vloterra FDEs(VFDEs), especially for delay differential equations(DDEs), has been widely discussed by many authors and a great deal of results have been found. As to the linear stability analysis of numerical methods for DDEs, we refer to the works of Barwell, Watanabe, Zennaro, Spijker, in't Hout, Bellen, Jackiewicz, Liu mingzhu, Kuang jiaoxun, Tian hongjiong, Zhang chengjian, Hu guangda and so on. The main results can be found in the monograph of Bellen and Zennaro or Kuang. Nonlinear stability analysis of numerical methods for DDEs originated with Torelli(in 1989) and Bellen and Zennare(in 1992). In 1999, Huang and Li et al. pointed out that the requirements of Torelli's stability are so strong that only a few methods with low order satisfy the conditions and put forward a newly reasonable stability concept. On the basis of the new concept, the study of nonlinear numerical stabilty for DDEs has been developed vigorously. Though the study limited to constant delay, fixed step, linear interpolation and negative Lipschitz constant at that time, the research object contained almost the commonly used numerical methods for DDEs and a great deal of new stability results had been obtained. Convergence of numerical methods for DDEs is another important issue and lots of results based on the classical Lipschitz condition can be found. As to the case of VFDEs, we refer to the momograph of Li(in 1997). For the case of DDEs, we refer to the papers of Oberle, Pesch, Bellen, Zennaro, Tavernini, Arndt, Enright, Feldstein, Neves, Karoui, Vaillancourt, Baker and Paul etc. However, the above results are only suitable to nonstiff DDEs, however, which are not suitable to stiff DDEs. Convergence analysis of numerical methods for stiff DDEs originated from the works of Zhang et al. in 1997. They introduced the concept of D-convergence and proved that several implicit
Functional differential equations(FDEs) arise widely in the fields of control theory, biology, medicine, chemistry, economics and so on. It is meaningful to investigate the theory and application of numerical methods for FDEs. In recent 30 years, the theory of computational methods for Vloterra FDEs(VFDEs), especially for delay differential equations(DDEs), has been widely discussed by many authors and a great deal of results have been found. As to the linear stability analysis of numerical methods for DDEs, we refer to the works of Barwell, Watanabe, Zennaro, Spijker, in't Hout, Bellen, Jackiewicz, Liu mingzhu, Kuang jiaoxun, Tian hongjiong, Zhang chengjian, Hu guangda and so on. The main results can be found in the monograph of Bellen and Zennaro or Kuang. Nonlinear stability analysis of numerical methods for DDEs originated with Torelli(in 1989) and Bellen and Zennare(in 1992). In 1999, Huang and Li et al. pointed out that the requirements of Torelli's stability are so strong that only a few methods with low order satisfy the conditions and put forward a newly reasonable stability concept. On the basis of the new concept, the study of nonlinear numerical stabilty for DDEs has been developed vigorously. Though the study limited to constant delay, fixed step, linear interpolation and negative Lipschitz constant at that time, the research object contained almost the commonly used numerical methods for DDEs and a great deal of new stability results had been obtained. Convergence of numerical methods for DDEs is another important issue and lots of results based on the classical Lipschitz condition can be found. As to the case of VFDEs, we refer to the momograph of Li(in 1997). For the case of DDEs, we refer to the papers of Oberle, Pesch, Bellen, Zennaro, Tavernini, Arndt, Enright, Feldstein, Neves, Karoui, Vaillancourt, Baker and Paul etc. However, the above results are only suitable to nonstiff DDEs, however, which are not suitable to stiff DDEs. Convergence analysis of numerical methods for stiff DDEs originated from the works of Zhang et al. in 1997. They introduced the concept of D-convergence and proved that several implicit
引文
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