刚性微分方程几类高效数值方法及中立型泛函微分方程数值稳定性分析
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摘要
刚性微分方程常出现于航空、航天、热核反应、自动控制、电子网络及化学动力学等一系列高科技领域,其数值解法具有毋庸质疑的重要性。偏微分方程初边值问题半离散化而获得的大规模常微分方程组是产生刚性的另一个重要源泉。近几十年来,刚性微分方程算法理论获得了大量重要成果。对于刚性问题的计算,要求数值方法的绝对稳定域尽可能的大;同时要求Stiff分量的数值误差能够迅速衰减,从而要求方法在∞点是极端稳定的。因此构造这两方面都具有优势的高效数值方法一直是刚性问题研究的重要课题之一。
一些著名的数值方法如BDF方法等在∞点是极端稳定的,但高阶方法的稳定域不够理想;而Gauss型Runge-Kutta方法尽管是A-稳定的,但在∞点不是强稳定的。为此,构造在∞点是极端稳定的且有较大稳定域的高效算法是本文的第一项工作:
(1)构造了3-6阶改进的向后微分公式IBDF1及3-7阶改进的向后微分公式IBDF2;
(2)构造了5-9阶改进的Enright方法;
以上三类新的方法分别保持了BDF方法和Enright方法固有优点,但其稳定域得到了较大改善,有较好的应用前景。
(3)运用模式搜索法得到了几类在∞点稳定性最优的s级r步Gauss型多步Runge-Kutta方法,其中s=1,2,3,r=2,3,这些方法在∞点的稳定性远优于s级Gauss型单步Runge-Kutta方法。理论分析和数值试验表明,对于求解强刚性问题,前者的实际计算精度远高于后者。
应该注意到在∞点L-稳定的Gauss型多步Runge-Kutta方法是不存在的,能获得使稳定矩阵在∞处的谱半径最小的强稳定的s级r步Gauss型多步Runge-Kutta方法对于强刚性问题的求解是非常有利的。
中立型泛函微分方程(NFDEs)常出现于生物学、物理学、控制理论及工程技术等诸多领域。在过去的几十年里,许多学者致力于数值方法的线性稳定性研究并获得了大量重要成果。最近,一些学者就非线性中立型延迟微分方程(NDDEs)和非线性中立型延迟积分微分方程(NDIDEs)的数值稳定性进行了研究。在此基础上进一步研究中立型延迟积分微分方程及更一般的中立型泛函微分方程的数值稳定性是本文的另一项工作:
(4)研究了中立型延迟积分微分方程线性多步法的数值稳定性。结果表明:在问题本身渐近稳定的条件下,A-稳定的线性多步法也是渐近稳定的。
(5)研究了巴拿赫空间中中立型泛函微分方程显式和对角隐式Rung-Kutta方法的非线性稳定性。获得了一些显式和对角隐式Rung-Kutta方法用于求解非线性中立型泛函微分方程时的数值稳定性和条件收缩性结果,数值试验进一步检验了这些理论结果的正确性。
应当指出,在国内外其他文献中,迄今主要研究了中立型延迟微分方程数值方法的稳定性,尚未见到关于Banach空间中一般的非线性中立型泛函微分方程数值稳定性的研究工作。
Stiff differential equations can be found in the high-tech fields such as aviation, spaceflight, thermonuclear reaction, automatic control, electronic network and chemical kinetics and so on. The numerical methods for these stiff problems are undoubtedly important. The large-scale ordinary differential equations derived from the semi-discretization of partial differential equations are another important sources of stiff differential equations. In the last few decades, a lots of important results on the theory of computational methods for stiff differential equations have been obtained. The absolute stability regions are required as big as possible for computing the stiff problems and the numerical methods are required to be extremely stable at∞. Therefore the construction of highly efficient numerical methods which have advantages on both sides is always one of important research subject of stiff problems.
Some famous numerical methods such as BDF and etc are extremely stable at∞. But the stability regions of high-order methods are not ideal enough. Although Runge-Kutta methods of Gauss type are A-stable, they are not strongly stable at∞. So the construction of highly efficient algorithm which has strong stability at∞and bigger stability region is the first work of this dissertation:
(1) Two classes of improved backward differentiation formulae are presented, whose abbreviation are IBDF1 and IBDF2 respectively;
(2) The improved Enright methods with order 5-9 are presented;
The above new methods preserve the original advantages of BDF and Enright methods respectively and their stability regions have been achieved some large improvements. So the application prospect is very extensive.
(3) Several classes of multistep Runge-Kutta methods of Gauss type which have the optimal stability at∞are obtained by using pattern search method. The stability at∞of these new methods is superior to the one-step Runge-Kutta methods of Gauss type. Theoretical analysis and numerical experiments show that the former's actual calculation accuracy are far higher than the latter for computing strongly stiff problems.
It is should noticed that there does not exist the multistep Runge-Kutta method of Gauss type which are L-stable at∞. It is beneficial to obtain these strongly stable methods which stability matrix's spectral radius is minimal at∞.
The Neutral Functional Differential Equations (NFDEs) often arise in biology, physics, control theory, engineering technology and so on. In the last few years, many authors have investigated the linear stability of numerical methods and obtained a lot of important results. Recently, some authors have studied the numerical stability of nonlinear neutral delay differential equations (NDDEs) and neutral delay integro-differential equations (NDIDEs). The further research of numerical stability for NDIDEs and more general NFDEs on basis of these results is an another work of this dissertation:
(4) The numerical stability results of linear multistep methods for NDIDEs are obtained. Theoretical analysis shows that A-stable linear multistep methods are also asymptotically stable if the problems are asymptotically stable.
(5) The nonlinear stability of explicit and diagonally implicit Runge-Kutta methods for neutral functional differential equations (NFDEs) are discussed in Banach spaces. The results on the numerical stability and conditional contractibility of some explicit and diagonally implicit Runge-Kutta methods for nonlinear NFDEs are obtained. Numerical examples are given to confirm the theoretical results.
It should be pointed out that the other related literatures of home and abroad mainly discussed the stability of numerical methods for NDDEs. The research of numerical stabilities of general nonlinear NFDEs in Banach space haven't been seen yet.
一些著名的数值方法如BDF方法等在∞点是极端稳定的,但高阶方法的稳定域不够理想;而Gauss型Runge-Kutta方法尽管是A-稳定的,但在∞点不是强稳定的。为此,构造在∞点是极端稳定的且有较大稳定域的高效算法是本文的第一项工作:
(1)构造了3-6阶改进的向后微分公式IBDF1及3-7阶改进的向后微分公式IBDF2;
(2)构造了5-9阶改进的Enright方法;
以上三类新的方法分别保持了BDF方法和Enright方法固有优点,但其稳定域得到了较大改善,有较好的应用前景。
(3)运用模式搜索法得到了几类在∞点稳定性最优的s级r步Gauss型多步Runge-Kutta方法,其中s=1,2,3,r=2,3,这些方法在∞点的稳定性远优于s级Gauss型单步Runge-Kutta方法。理论分析和数值试验表明,对于求解强刚性问题,前者的实际计算精度远高于后者。
应该注意到在∞点L-稳定的Gauss型多步Runge-Kutta方法是不存在的,能获得使稳定矩阵在∞处的谱半径最小的强稳定的s级r步Gauss型多步Runge-Kutta方法对于强刚性问题的求解是非常有利的。
中立型泛函微分方程(NFDEs)常出现于生物学、物理学、控制理论及工程技术等诸多领域。在过去的几十年里,许多学者致力于数值方法的线性稳定性研究并获得了大量重要成果。最近,一些学者就非线性中立型延迟微分方程(NDDEs)和非线性中立型延迟积分微分方程(NDIDEs)的数值稳定性进行了研究。在此基础上进一步研究中立型延迟积分微分方程及更一般的中立型泛函微分方程的数值稳定性是本文的另一项工作:
(4)研究了中立型延迟积分微分方程线性多步法的数值稳定性。结果表明:在问题本身渐近稳定的条件下,A-稳定的线性多步法也是渐近稳定的。
(5)研究了巴拿赫空间中中立型泛函微分方程显式和对角隐式Rung-Kutta方法的非线性稳定性。获得了一些显式和对角隐式Rung-Kutta方法用于求解非线性中立型泛函微分方程时的数值稳定性和条件收缩性结果,数值试验进一步检验了这些理论结果的正确性。
应当指出,在国内外其他文献中,迄今主要研究了中立型延迟微分方程数值方法的稳定性,尚未见到关于Banach空间中一般的非线性中立型泛函微分方程数值稳定性的研究工作。
Stiff differential equations can be found in the high-tech fields such as aviation, spaceflight, thermonuclear reaction, automatic control, electronic network and chemical kinetics and so on. The numerical methods for these stiff problems are undoubtedly important. The large-scale ordinary differential equations derived from the semi-discretization of partial differential equations are another important sources of stiff differential equations. In the last few decades, a lots of important results on the theory of computational methods for stiff differential equations have been obtained. The absolute stability regions are required as big as possible for computing the stiff problems and the numerical methods are required to be extremely stable at∞. Therefore the construction of highly efficient numerical methods which have advantages on both sides is always one of important research subject of stiff problems.
Some famous numerical methods such as BDF and etc are extremely stable at∞. But the stability regions of high-order methods are not ideal enough. Although Runge-Kutta methods of Gauss type are A-stable, they are not strongly stable at∞. So the construction of highly efficient algorithm which has strong stability at∞and bigger stability region is the first work of this dissertation:
(1) Two classes of improved backward differentiation formulae are presented, whose abbreviation are IBDF1 and IBDF2 respectively;
(2) The improved Enright methods with order 5-9 are presented;
The above new methods preserve the original advantages of BDF and Enright methods respectively and their stability regions have been achieved some large improvements. So the application prospect is very extensive.
(3) Several classes of multistep Runge-Kutta methods of Gauss type which have the optimal stability at∞are obtained by using pattern search method. The stability at∞of these new methods is superior to the one-step Runge-Kutta methods of Gauss type. Theoretical analysis and numerical experiments show that the former's actual calculation accuracy are far higher than the latter for computing strongly stiff problems.
It is should noticed that there does not exist the multistep Runge-Kutta method of Gauss type which are L-stable at∞. It is beneficial to obtain these strongly stable methods which stability matrix's spectral radius is minimal at∞.
The Neutral Functional Differential Equations (NFDEs) often arise in biology, physics, control theory, engineering technology and so on. In the last few years, many authors have investigated the linear stability of numerical methods and obtained a lot of important results. Recently, some authors have studied the numerical stability of nonlinear neutral delay differential equations (NDDEs) and neutral delay integro-differential equations (NDIDEs). The further research of numerical stability for NDIDEs and more general NFDEs on basis of these results is an another work of this dissertation:
(4) The numerical stability results of linear multistep methods for NDIDEs are obtained. Theoretical analysis shows that A-stable linear multistep methods are also asymptotically stable if the problems are asymptotically stable.
(5) The nonlinear stability of explicit and diagonally implicit Runge-Kutta methods for neutral functional differential equations (NFDEs) are discussed in Banach spaces. The results on the numerical stability and conditional contractibility of some explicit and diagonally implicit Runge-Kutta methods for nonlinear NFDEs are obtained. Numerical examples are given to confirm the theoretical results.
It should be pointed out that the other related literatures of home and abroad mainly discussed the stability of numerical methods for NDDEs. The research of numerical stabilities of general nonlinear NFDEs in Banach space haven't been seen yet.
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