积分微分方程数值方法的散逸性
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摘要
在物理学和工程技术学中，许多的动力系统都具有散逸性，即系统具有一有界的吸引集，使从任意初始条件出发的解经过有限时间后进入并随后始终保持在这个吸引集里面。如二维的Navier-Stokes方程以及Lorenz等许多重要系统都是散逸的。
散逸性研究一直是动力系统研究中的重要课题(参见Teman)。当用数值方法求解这些系统时，自然希望数值方法能继承原系统所固有的散逸性。
积分微分方程广泛出现于物理、工程、生物、医学及经济等领域，其算法理论研究具有无容置疑的重要性，近年来逐渐引起众多学者的极大关注。对于积分微分方程甘四清[7]首先研究了该系统的散逸性及θ-方法的散逸性。
本文在文献[7]研究的基础上，进一步研究积分微分方程数值方法的散逸性。本文主要结果如下：
(1)当积分项用CQ公式逼近时，我们证明了(k，l)-代数稳定的Runge-Kutta方法当k≤1时是有限维散逸的，当k＜1时是无限维散逸的。
(2)当积分项用PQ公式逼近时，证明了(k，l)-代数稳定的Runge-Kutta方法当k≤1时是有限维散逸的。
(3)讨论了单支方法的散逸性，证明了G(c，p，O)-代数稳定的单支方法(积分项用CQ公式逼近)当c≤1时是有限维散逸的，当c＜1时则是无限维散逸的。
(4)讨论了一类线性多步法的散逸性，给出了该方法(积分项用CQ公式逼近)散逸的充分条件。
(5)对多步Runge-Kutta方法的散逸性进行了研究，当积分项用CQ公式逼近时，给出了该方法是有限维及无限维散逸的充分条件。
(6)通过数值试验，对Runge-Kutta方法，单支方法以及线性多步法的散逸性进行了测试，测试结果进一步验证了本文所获理论结果的正确性。
Many dynamical systems in physics and engineering are dissipative. These systems are characterized by possessing a bounded absorbing set which all trajectories enter in a finite time and thereafter remain inside. For example, the two-dimentional Navier-Stokes equation and other important systems such as Lorenz are dissipative.
The research of the disssipativity has always been important topic in the dynamical systems research (see Teman). When considering the applicability of numerical methods for these systems, it is important to analyze whether or not numerical methods inherit the dissipativity of the underlying systems.
Integro-differential equations (IDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Economics and so on. The theory of computational methods has decisive importance in Numerical Integro-differential equations. Recently, many scholars have pay careful attention to it. For the Integro-differential equation S.Gan first studied the dissipativity of the system and theθ-methods applied to the above system.
The thesis is concerned with the dissipativity of numerical methods for the underlying system on the basis of the study of S.Gan. Our main results in the thesis are as follows:
(1) When the integration term is approximated by the CQ formula, it is proved that the (k,l)-algebraically stable Runge-Kutta methods are dissipative in finite-dimentional space for k≤1 and are dissipative in infinite- dimentional space for k＜1.
(2) When the integration term is approximated by PQ formula, we proved that the (k,l)-algebraically stable methods are dissipative in finite-dimentional space for k≤1.
(3) We considered the dissipativity of the one-leg methods. When the integration term is approximated by the CQ formula, we proved that the G(c,p,0)-algebraically stable one-leg methods are dissipative in finite-dimentional space for k≤1 and are dissipative in infinitedimentional space for k＜1.
(4) We studied the dissipativity of the linear multistep methods. A sufficient condition for the dissipativity of the methods (the integration term is approximated by the CQ formula) is given.
(5) We investigated the dissipativity for the multistep Runge-Kutta methods when the integration term is approximated by the CQ formula. The sufficient conditions for the dissipativity of the methods in finite-dimentional and infinite-dimentional space are given.
(6) Numerical experiments are given for checking the dissipativity of Runge-Kutta methods, one-leg methods and linear multistep methods, which confirm the theoretical results obtained in this paper.
散逸性研究一直是动力系统研究中的重要课题(参见Teman)。当用数值方法求解这些系统时，自然希望数值方法能继承原系统所固有的散逸性。
积分微分方程广泛出现于物理、工程、生物、医学及经济等领域，其算法理论研究具有无容置疑的重要性，近年来逐渐引起众多学者的极大关注。对于积分微分方程甘四清[7]首先研究了该系统的散逸性及θ-方法的散逸性。
本文在文献[7]研究的基础上，进一步研究积分微分方程数值方法的散逸性。本文主要结果如下：
(1)当积分项用CQ公式逼近时，我们证明了(k，l)-代数稳定的Runge-Kutta方法当k≤1时是有限维散逸的，当k＜1时是无限维散逸的。
(2)当积分项用PQ公式逼近时，证明了(k，l)-代数稳定的Runge-Kutta方法当k≤1时是有限维散逸的。
(3)讨论了单支方法的散逸性，证明了G(c，p，O)-代数稳定的单支方法(积分项用CQ公式逼近)当c≤1时是有限维散逸的，当c＜1时则是无限维散逸的。
(4)讨论了一类线性多步法的散逸性，给出了该方法(积分项用CQ公式逼近)散逸的充分条件。
(5)对多步Runge-Kutta方法的散逸性进行了研究，当积分项用CQ公式逼近时，给出了该方法是有限维及无限维散逸的充分条件。
(6)通过数值试验，对Runge-Kutta方法，单支方法以及线性多步法的散逸性进行了测试，测试结果进一步验证了本文所获理论结果的正确性。
Many dynamical systems in physics and engineering are dissipative. These systems are characterized by possessing a bounded absorbing set which all trajectories enter in a finite time and thereafter remain inside. For example, the two-dimentional Navier-Stokes equation and other important systems such as Lorenz are dissipative.
The research of the disssipativity has always been important topic in the dynamical systems research (see Teman). When considering the applicability of numerical methods for these systems, it is important to analyze whether or not numerical methods inherit the dissipativity of the underlying systems.
Integro-differential equations (IDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Economics and so on. The theory of computational methods has decisive importance in Numerical Integro-differential equations. Recently, many scholars have pay careful attention to it. For the Integro-differential equation S.Gan first studied the dissipativity of the system and theθ-methods applied to the above system.
The thesis is concerned with the dissipativity of numerical methods for the underlying system on the basis of the study of S.Gan. Our main results in the thesis are as follows:
(1) When the integration term is approximated by the CQ formula, it is proved that the (k,l)-algebraically stable Runge-Kutta methods are dissipative in finite-dimentional space for k≤1 and are dissipative in infinite- dimentional space for k＜1.
(2) When the integration term is approximated by PQ formula, we proved that the (k,l)-algebraically stable methods are dissipative in finite-dimentional space for k≤1.
(3) We considered the dissipativity of the one-leg methods. When the integration term is approximated by the CQ formula, we proved that the G(c,p,0)-algebraically stable one-leg methods are dissipative in finite-dimentional space for k≤1 and are dissipative in infinitedimentional space for k＜1.
(4) We studied the dissipativity of the linear multistep methods. A sufficient condition for the dissipativity of the methods (the integration term is approximated by the CQ formula) is given.
(5) We investigated the dissipativity for the multistep Runge-Kutta methods when the integration term is approximated by the CQ formula. The sufficient conditions for the dissipativity of the methods in finite-dimentional and infinite-dimentional space are given.
(6) Numerical experiments are given for checking the dissipativity of Runge-Kutta methods, one-leg methods and linear multistep methods, which confirm the theoretical results obtained in this paper.
引文
[1] C. T. H. Baker and N. J. Ford, Stability properties of a scheme for the approximation solution of a delay integro-differential equtions, Appl. Numer. Math., 9(1992), 357-370.
[2] H. Brunner, The numerical solution of neutral Volterra integro-differentianl equtiona with delay argument. Ann. Numer., 1(1994), 309-322.
[3] K. Burrage and J. C. Butcher, Nonlinear stability of a general class of differential eqution methods, BIT., 20(1980), 185-203.
[4] J. C. Butcher, A stability property of implicit Runge-Kutta Methods. BIT., 15(1975), 358-361.
[5] G. Dahquist, A special stability problem for liner multistep meshods, BIT., 3(1963), 358-361.
[6] G. Dahquist, Error analysis for a class of methods for stiff nonlinear initial value problems, Num. Anal(Dundee 1975) (Springer Verlag in mathmatics 506). Berlin: Springer, 60-74.
[7] S. Gan, Disssipativity of linear θ-methods for integro-differential equations, (to be appeared in Computers and Mathematics with Applications).
[8] A. T. Hill, Dissipativity of Runge-Kutta Methods in Hilbert spaces, BIT., 37(1997a), 37-42.
[9] A. T. Hill, Global dissipativity for A-Stable methods, SIAM J. Numer. Anal., 34(1997b), 119-142.
[10] C. Huang, Disssipativity of Runge-Kutta Methods for dissipative systems with delays, IMAJ. Numer. Anal., 20(2000), 153-166.
[11] A. R. Humphries and A. M. Stuart, Runge-Kutta Methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal., 31(1994), 1452-1485.
[12] A. R. Humphries and A. M. Stuart, Model problems in numerical stability theory for initial value problems, SIAM Review., 36(1994), 226-257.
[13] 黄乘明，非线性延迟微分方程线性多步方法的收缩性，湘潭大学自然科学学报，21：3(1999)，4-6．
[14] C. Huang, H. Fu, S. Li and G. Chen, Stability analysis of Runge-Kutta methods for non-linear delay differential equations, BIT., 39(1999), 270-280.
[15] C. Huang, Dissipativity of one-leg methods for dynamical systems with delays, J. Comput. Appl. Math., 25(1999), 15-26.
[16] C. Huang, Dissipativity of Multistep Runge-Kutta Method for dynamical systems with delays, Math and Comput Modelling., 40(2004), 1285-1296.
[17] T. Koto, Stability of Runge-Kutta Methods for delay integro-differential equations, J. Comput. Appl. Math., 145(2002), 483-492..
[18] T. Koto, Stability of θ-methods for delay integro-differential equations, J. Comput. Appl. Math., 161(2003), 393-404.
[19] 李寿佛，刚性微分方程的算法理论，湖南科技出版社(长沙)，1997年．
[20] H. Tian, Numerical and analytic dissipativity of the θ-method for delay differential eqution with a bounded leg, Int. J. Bifurcation and Chaos., 14(2004), 1839-1845.
[21] Torelli. Stability of numerical methods for delay differencial equations. J. Comput. Appl. Math., 25(1989), 15-26.
[22] 文立平，余越昕，李寿佛，一类求解分片延迟微分方程的线性多步法的散逸性，计算数学，28：1(2006)，67-74．
[23] 文立平，抽象空间中非线性Volterra泛涵微分方程的数值稳定性分析，湘潭大学博士论文，2005年．
[24] 肖爱国，Hilbert空间中散逸动力系统的一般线性方法的散逸稳定性，计算数学，22：4(2000)，501-506．
[25] A. Xiao, On the solvability of general linear methods for dissipative dynamical systems, J. Comput. Appl. Math., 18(2000), 633-638.
[26] C. Zhang, Stability analysis of Runge-Kutta Methods for nonlinear Volterra delay integro-differential equations IMA Numer. Anal., 24(2004), 193-214.
[27] 张长海，梁久祯，刘明珠，分片延迟微分方程θ—方法数值解渐近稳定性，数值计算与计算机应用，21：4(2000)，241—246．
[28] A. Bellen and M. Zennaro, Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math. 9(1992), 321-346.
[29] E. Hairer and G. Wanner, Solving ordinary differential equations Ⅱ, Berlin: Springer(1991).
[30] R. Teman, Infinite-dimentional dynamical systems in mechanics and physics, Springer Applied Mathmatical Science Series 68, Berlin: Springer(1988).
[31] M. Crouzeix, Sur la B-stability des methods de Runge-Kutta, Numer. Math., 32(1979), 75-82.
[32] K. Dekker and J. G. Verwen, Stability of Runge-Kutta methods for stiff nonlinear differential equations(CWI Monographs 2), Amsterdan: North-Holland, (1984).
[33] Y. A. Bickart, P-stable and P[α, β]-stable intergration/interpolation methods in the solution of retarded differential-difference equations, BIT., 22(1982), 464-476.
[34] K. J. In't, A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT., 32, 634-649.
[35] D. S. Watanabe and M. G. Roth, The stability of difference formulas for delay differential equations, SIAM. Numer. Anal., 22(1985), 132-145.
[36] M. Zennaro, P-stability of Runge-Kutta methods for delay differential equations, Numer. Math., 49(1986), 305-345.
[37] 肖爱国，一般线性方法的散逸稳定性，高等学校计算数学学报，第二期，(1996)，183-189．
[38] S. Gan, Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations,(to appear in Comput. Math. Appl).
[39] S. Gan, Exact and discretized dissipativity of the pantograph equation, (to appear in J. Comput. Math).
[40] J. K. Hale, Homoclinic orbits and chaos in delay equations, In proceedings of the Ninth Dundee Conference on Ordinary and Partial Differential Equations, Edited by B. D. Sleeman and R. J. Jarvis, John Wiley and Sons, New York,(1986).
[41] S. Li, Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach space, Science in China Ser. A. Math., 48(2005), 372-387.
[42] C. Zhang and S. Li, Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional differential equations, Comput. Math, appl., (2000), 109-115.
[43] 黄乘明，陈光南，延迟动力系统线性θ-方法的散逸性，计算数学，22：4(2000)，501-506．
[44] 张长海，宋久祯，刘明珠，分片延迟微分方程θ-方法数值解的渐进稳定性，数值计算与计算机应用，2：14(2000)，241-246．
[45] K. Lorenz, Determinitic nonperiodic flow, J. Atmospheric Sci., 20(1963), 130-141.
[46] L. Wen and S. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324(2006), 696-706.
[47] V. K. Barwell, Special stability problems for functional differential equations, BIT., 15(1975), 130-1359.
[48] C. Huang, H. Y. Fu, S. Li and G. Chen, Stability analysis of Runge-Kutta methods for non-linear delay differential equations, BIT., 39(1999), 270-280.
[49] C. Huang, S. Li, H. Fu and G. Chen, Stability and error analysis of one-leg methods for nonlinear delay differential equations, J. Comput. Appl. Math., 103(1999), 267-279.
[50] D. S. Watanabe and M. G. Roth, The stability of difference formulas for delay differencial equations, SIAM J. Numer. Anal., 22(1985), 132-145.
[51] M. Zennaro, On the P-stability of one-leg collocation for delay differential equation, ISNM., 74(1985), 334-343.
[52] M. Liu and M. N. Spijker, The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. Numer. Anal., 10(1990), 31-48.
[53] M. Zennaro, P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math., 49(1986), 305-318.
[54] K. J. in't Hout, The stability of θ-methods for systems of delay differential equations, Ann. Numer. Math., 1(1994), 323-334.
[55] K. J. in't Hout, Stability analysis of Runge-Kutta methods for delay differential equations, IMA J. Numer. Anal., 17(1997), 17-27.
[56] T. Koto, A stability properties of A-stable natural Runge-Kutta methods for systems of delay differential equations, BIT., 34(1994), 262-267.
[57] H. Tian and J. Kuang, The stability of linear multistep methods for systems of delay differential equations, Numer. Math. A Journal of Chinese Universities, 4(1995), 10-16.
[58] 张诚坚，泛函微分方程数值解的稳定性与 D-收敛性，湖南大学博士论文，1998年．
[59] C. Zhang and S. Vandewalle, Stability analysis of Volterra delay-integro-differential equations and the backward differention time discretization, J. Comput. Appl. Math., (2004) 164-165, 797-814.
[60] 余越昕，李寿佛，多延迟微分方程θ-方法的散逸性，湘潭师范学院学报，29：3(2001)，1-5．
[61] G. Dalquist and Jeltsch, Generalized disks of contractivity for explicit and implicit Runge-Kutta methods, Trita-Na report 7906(1979).
[2] H. Brunner, The numerical solution of neutral Volterra integro-differentianl equtiona with delay argument. Ann. Numer., 1(1994), 309-322.
[3] K. Burrage and J. C. Butcher, Nonlinear stability of a general class of differential eqution methods, BIT., 20(1980), 185-203.
[4] J. C. Butcher, A stability property of implicit Runge-Kutta Methods. BIT., 15(1975), 358-361.
[5] G. Dahquist, A special stability problem for liner multistep meshods, BIT., 3(1963), 358-361.
[6] G. Dahquist, Error analysis for a class of methods for stiff nonlinear initial value problems, Num. Anal(Dundee 1975) (Springer Verlag in mathmatics 506). Berlin: Springer, 60-74.
[7] S. Gan, Disssipativity of linear θ-methods for integro-differential equations, (to be appeared in Computers and Mathematics with Applications).
[8] A. T. Hill, Dissipativity of Runge-Kutta Methods in Hilbert spaces, BIT., 37(1997a), 37-42.
[9] A. T. Hill, Global dissipativity for A-Stable methods, SIAM J. Numer. Anal., 34(1997b), 119-142.
[10] C. Huang, Disssipativity of Runge-Kutta Methods for dissipative systems with delays, IMAJ. Numer. Anal., 20(2000), 153-166.
[11] A. R. Humphries and A. M. Stuart, Runge-Kutta Methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal., 31(1994), 1452-1485.
[12] A. R. Humphries and A. M. Stuart, Model problems in numerical stability theory for initial value problems, SIAM Review., 36(1994), 226-257.
[13] 黄乘明，非线性延迟微分方程线性多步方法的收缩性，湘潭大学自然科学学报，21：3(1999)，4-6．
[14] C. Huang, H. Fu, S. Li and G. Chen, Stability analysis of Runge-Kutta methods for non-linear delay differential equations, BIT., 39(1999), 270-280.
[15] C. Huang, Dissipativity of one-leg methods for dynamical systems with delays, J. Comput. Appl. Math., 25(1999), 15-26.
[16] C. Huang, Dissipativity of Multistep Runge-Kutta Method for dynamical systems with delays, Math and Comput Modelling., 40(2004), 1285-1296.
[17] T. Koto, Stability of Runge-Kutta Methods for delay integro-differential equations, J. Comput. Appl. Math., 145(2002), 483-492..
[18] T. Koto, Stability of θ-methods for delay integro-differential equations, J. Comput. Appl. Math., 161(2003), 393-404.
[19] 李寿佛，刚性微分方程的算法理论，湖南科技出版社(长沙)，1997年．
[20] H. Tian, Numerical and analytic dissipativity of the θ-method for delay differential eqution with a bounded leg, Int. J. Bifurcation and Chaos., 14(2004), 1839-1845.
[21] Torelli. Stability of numerical methods for delay differencial equations. J. Comput. Appl. Math., 25(1989), 15-26.
[22] 文立平，余越昕，李寿佛，一类求解分片延迟微分方程的线性多步法的散逸性，计算数学，28：1(2006)，67-74．
[23] 文立平，抽象空间中非线性Volterra泛涵微分方程的数值稳定性分析，湘潭大学博士论文，2005年．
[24] 肖爱国，Hilbert空间中散逸动力系统的一般线性方法的散逸稳定性，计算数学，22：4(2000)，501-506．
[25] A. Xiao, On the solvability of general linear methods for dissipative dynamical systems, J. Comput. Appl. Math., 18(2000), 633-638.
[26] C. Zhang, Stability analysis of Runge-Kutta Methods for nonlinear Volterra delay integro-differential equations IMA Numer. Anal., 24(2004), 193-214.
[27] 张长海，梁久祯，刘明珠，分片延迟微分方程θ—方法数值解渐近稳定性，数值计算与计算机应用，21：4(2000)，241—246．
[28] A. Bellen and M. Zennaro, Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math. 9(1992), 321-346.
[29] E. Hairer and G. Wanner, Solving ordinary differential equations Ⅱ, Berlin: Springer(1991).
[30] R. Teman, Infinite-dimentional dynamical systems in mechanics and physics, Springer Applied Mathmatical Science Series 68, Berlin: Springer(1988).
[31] M. Crouzeix, Sur la B-stability des methods de Runge-Kutta, Numer. Math., 32(1979), 75-82.
[32] K. Dekker and J. G. Verwen, Stability of Runge-Kutta methods for stiff nonlinear differential equations(CWI Monographs 2), Amsterdan: North-Holland, (1984).
[33] Y. A. Bickart, P-stable and P[α, β]-stable intergration/interpolation methods in the solution of retarded differential-difference equations, BIT., 22(1982), 464-476.
[34] K. J. In't, A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT., 32, 634-649.
[35] D. S. Watanabe and M. G. Roth, The stability of difference formulas for delay differential equations, SIAM. Numer. Anal., 22(1985), 132-145.
[36] M. Zennaro, P-stability of Runge-Kutta methods for delay differential equations, Numer. Math., 49(1986), 305-345.
[37] 肖爱国，一般线性方法的散逸稳定性，高等学校计算数学学报，第二期，(1996)，183-189．
[38] S. Gan, Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations,(to appear in Comput. Math. Appl).
[39] S. Gan, Exact and discretized dissipativity of the pantograph equation, (to appear in J. Comput. Math).
[40] J. K. Hale, Homoclinic orbits and chaos in delay equations, In proceedings of the Ninth Dundee Conference on Ordinary and Partial Differential Equations, Edited by B. D. Sleeman and R. J. Jarvis, John Wiley and Sons, New York,(1986).
[41] S. Li, Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach space, Science in China Ser. A. Math., 48(2005), 372-387.
[42] C. Zhang and S. Li, Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional differential equations, Comput. Math, appl., (2000), 109-115.
[43] 黄乘明，陈光南，延迟动力系统线性θ-方法的散逸性，计算数学，22：4(2000)，501-506．
[44] 张长海，宋久祯，刘明珠，分片延迟微分方程θ-方法数值解的渐进稳定性，数值计算与计算机应用，2：14(2000)，241-246．
[45] K. Lorenz, Determinitic nonperiodic flow, J. Atmospheric Sci., 20(1963), 130-141.
[46] L. Wen and S. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324(2006), 696-706.
[47] V. K. Barwell, Special stability problems for functional differential equations, BIT., 15(1975), 130-1359.
[48] C. Huang, H. Y. Fu, S. Li and G. Chen, Stability analysis of Runge-Kutta methods for non-linear delay differential equations, BIT., 39(1999), 270-280.
[49] C. Huang, S. Li, H. Fu and G. Chen, Stability and error analysis of one-leg methods for nonlinear delay differential equations, J. Comput. Appl. Math., 103(1999), 267-279.
[50] D. S. Watanabe and M. G. Roth, The stability of difference formulas for delay differencial equations, SIAM J. Numer. Anal., 22(1985), 132-145.
[51] M. Zennaro, On the P-stability of one-leg collocation for delay differential equation, ISNM., 74(1985), 334-343.
[52] M. Liu and M. N. Spijker, The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. Numer. Anal., 10(1990), 31-48.
[53] M. Zennaro, P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math., 49(1986), 305-318.
[54] K. J. in't Hout, The stability of θ-methods for systems of delay differential equations, Ann. Numer. Math., 1(1994), 323-334.
[55] K. J. in't Hout, Stability analysis of Runge-Kutta methods for delay differential equations, IMA J. Numer. Anal., 17(1997), 17-27.
[56] T. Koto, A stability properties of A-stable natural Runge-Kutta methods for systems of delay differential equations, BIT., 34(1994), 262-267.
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