Volterra泛函微分方程数值方法的稳定性分析
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摘要
本文将常微分方程初值问题类K_(σ,τ)(李寿佛1987年提出)及相应的延迟微分方程初值问题类(黄乘明2002年BIT)进一步拓广到一般的Volterra泛函微分方程初值问题类K_(σ,τ,β)：其中f：[0，T]×R~m×C_m[-d，T]→R~m满足条件：
将(θ，p，q)-代数稳定的Runge-Kutta方法及(k，p，q)-代数稳定的一般线性方法应用于K_(σ,τ,β)类问题，获得了这些方法的稳定性结果．这些结果可视为常微分方程及延迟微分方程相应数值方法稳定性结果的推广和发展，也可视为Volterra泛函微分方程相应结果(李寿佛2003年和2005年获得)的进一步推广．即使对于延迟微分方程这种特殊的泛函微分方程，我们的结果比文献中已有的结果更加广泛和深刻．
In this paper, we generalize the initial value problem class of ordinary differentialequations K_(σ,τ)(given by S.F. Li in 1987) and corresponding problem class of delay differential equations (given by CM. Huang in 2002 BIT) to the general initial value problem class of Volterra functional differential equations K_(σ,τ,β)f:[0,T]×R~m×C_m[-d,T]→R~m satisfying the conditionSolving the problem class K_(σ,τ,β) by (θ,p, g)-algebraically stable Runge-Kutta methodsand (k, p,q)-algebraically stable general linear methods, we obtain the stability results of these methods. These results can be regarded as extension of the stabilityresults of corresponding numerical methods for ordinary differential equations and delay differential equations as well as extension of corresponding results for Volterra functional differential equations (obtained by S.F. Li in 2003 and 2005). Even though for delay differential equations which is special functional differential equations, our results are more comprehensive and profound than the results which have been existent.
将(θ，p，q)-代数稳定的Runge-Kutta方法及(k，p，q)-代数稳定的一般线性方法应用于K_(σ,τ,β)类问题，获得了这些方法的稳定性结果．这些结果可视为常微分方程及延迟微分方程相应数值方法稳定性结果的推广和发展，也可视为Volterra泛函微分方程相应结果(李寿佛2003年和2005年获得)的进一步推广．即使对于延迟微分方程这种特殊的泛函微分方程，我们的结果比文献中已有的结果更加广泛和深刻．
In this paper, we generalize the initial value problem class of ordinary differentialequations K_(σ,τ)(given by S.F. Li in 1987) and corresponding problem class of delay differential equations (given by CM. Huang in 2002 BIT) to the general initial value problem class of Volterra functional differential equations K_(σ,τ,β)f:[0,T]×R~m×C_m[-d,T]→R~m satisfying the conditionSolving the problem class K_(σ,τ,β) by (θ,p, g)-algebraically stable Runge-Kutta methodsand (k, p,q)-algebraically stable general linear methods, we obtain the stability results of these methods. These results can be regarded as extension of the stabilityresults of corresponding numerical methods for ordinary differential equations and delay differential equations as well as extension of corresponding results for Volterra functional differential equations (obtained by S.F. Li in 2003 and 2005). Even though for delay differential equations which is special functional differential equations, our results are more comprehensive and profound than the results which have been existent.
引文
[1]李寿佛．刚性微分方程算法理论[M].长沙：湖南科学技术出版社， 1997．
[2] A. Bellen and M. Zennaro. Numerical methods for delay differential equations[M]. Oxford University Press, Oxford, 2003.
[3] J. D. Lambert. Numerical analysis of ordinary differential equations[M]. Wiley, Chichester, 1991.
[4] J. C. Butcher. The numerical analysis of ordinary differential equations[M]. John Wiley, Chichester, New York, Berlin, 1993.
[5] E. Hairer, G. Wanner. Solving ordinary differential equations II: Stiff and Differential Algebraic Problems[M]. Springer, Berlin, 1993.
[6] C. T. H. Baker. Retarded differential equations[J]. J. Comput. Appl. Math. 2000, 125: 309-335.
[7] K.J.in't Hout, M.N.Spijker. Stability analysis of numerical methods for delay differential equations[J]. Numer.Math. 1991, 59: 807-814.
[8] L. Torelli. Stability of numerical methods for delay differential equations[J]. J. Comput. Appl. Math. 1989, 25: 15-26.
[9] T.Koto. A stability property of A-stable natural Runge-Kutta methods for systems of delay differential equations[J]. BIT. 1994, 34: 262-267.
[10] N.Guglielmi. Asymptotic stability barriers for natural Runge-Kutta processes for delay equations[J]. SIMA J.Numer.Anal. 2001, 39: 738-763.
[11] CM. Huang. Numerical analysis of nonlinear delay differential equations[C]. Ph.D. Thesis, China Academy of Engineering Physics, 1999.
[12] C. M. Huang, S. F. Li, H. Y. Hu and G. N. Chen. Nonlinear stability of general linear methods for delay differential equations[J]. BIT. 2002, 42: 380-392.
[13] C.M. Huang, H.Y. Fu.S.F. Li. G.N. Chen. Stability analysis of Runge-Kutta methods for nonlinear delay differential equations[J]. BIT. 1999, 39: 270-280.
[14] C.M. Huang, H.Y. Fu,S.F. Li, G.N. Chen. Stability and error analysis of one-leg methods for nonlinear delay differential equations[J]. J.Comput.Appl.Math.1999, 103: 263-279.
[15] C.J. Zhang, S.Z. Zhou. Nonlinear stability and D-convergence of Runge-Kutta methods for DDEs[J]. J.Comput.Appl.Math. 1997, 85: 225-237.
[16] M. Z. Liu and M. N. Spijker. The stability of the methods in the numerical solution of delay differential equations[J]. IMA J. Numer. Anal. 1990, 10: 31-48.
[17] S.F. Li. B-theory of Runge-Kutta methods for stiff Volterra functional differential equations[J]. SCIENCE IN CHINA (Series A). 2003, 46 (5): 662-674.
[18] S.F. Li. B-theory of general linear methods for stiff Volterra functional differential equations[J]. Appl.Numer. Math. 2005, 53 (1): 57-72.
[19] S.F. Li. Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach space[J]. SCIENCE IN CHINA (Series A).2005, 48 (3): 372-387.
[20] L.P. Wen, Y.X. Yu, S.F. Li. The test problem class of Volterra functional differential equations in Banach space[J]. Appl. Math.Comput. 2006, 179: 30-38.
[21] L.R Wen, Y.X. Yu, S.F. Li. Stability of explicit and diagonal implicit Runge-Kutta methods for nonlinear Volterra functional differential equations in Banach space[J]. Appl. Math.Comput. 2006, 183: 68-78.
[22] L.P. Wen, S.F. Li. Nonlinear stability of linear multistep methods for stiff delay differential equations in Banach space[J]. Appl. Math.Comput. 2005,168: 1031-1044.
[23]李寿佛．Runge-Kutta方法的稳定性准则[J]．湘潭大学自然科学学报．1987， 3：11-17．
[24] K. Burrage and J.C. Butcher. Non-linear stability of a general class of differentialequations methods[J]. BIT. 1980, 20: 185-203.
[25] K. Burrage and J.C. Butcher. Stability criteria for implicit Runge-Kutta methods[J]. SIAM J.Numer.Anal. 1979, 16: 46-57.
[26] J.C. Butcher. Linear and non-linear stability for general linear methods[J]. BIT. 1987, 27: 182-189.
[27] S.F. Li. Nonlinear stability of general linear methods[J]. J.Comput.Math. 1991, 9: 97-104.
[28]肖爱国． Runge-Kutta方法(θ，p，q)-代数稳定性[J].计算数学． 1993，4： 440-448．
[29] M.Zennaro. P-stability of Runge-Kutta methods for delay differential equations[J]. Numer.Math. 1986, 49: 305-318.
[30] M.Zennaro. Asymptotic stability analysis of Runge-Kutta methods for nonlinearsystems of delay differential equations[J]. Numer.Math. 1997, 77: 549-563.
[31] J.K. Hale. Homoclinic orbits and chaos in delay equations[M]. Proceedings of the Ninth Dundee Conference on Ordinary and Partial Differential Equations(B.D. Sleeman and R.J. Jarvis, eds). Wiley, New York, 1986.
[32] Gan Siqing, Zheng Weimin. Stability analysis of Runge-Kutta methods for delay integro-differential equations [J]. Tsinghua Science and Technology. 2004, 2: 185-188.
[33]甘四清．多步Runge-Kutta方法的收敛性与稳定性[J]．长沙铁道学院学报． 1997,3：62-66．
[34]李寿佛．一般线性方法的非线性稳定性[C]．全国首届高校常微分方程数值方 法研讨会．兰州，1987．
[35]肖飞雁，王文强．一类非线性多延迟微分方程一般线性方法的稳定性[J]．西北 师范大学学报(自然科学版)．2004，4：19-22．
[2] A. Bellen and M. Zennaro. Numerical methods for delay differential equations[M]. Oxford University Press, Oxford, 2003.
[3] J. D. Lambert. Numerical analysis of ordinary differential equations[M]. Wiley, Chichester, 1991.
[4] J. C. Butcher. The numerical analysis of ordinary differential equations[M]. John Wiley, Chichester, New York, Berlin, 1993.
[5] E. Hairer, G. Wanner. Solving ordinary differential equations II: Stiff and Differential Algebraic Problems[M]. Springer, Berlin, 1993.
[6] C. T. H. Baker. Retarded differential equations[J]. J. Comput. Appl. Math. 2000, 125: 309-335.
[7] K.J.in't Hout, M.N.Spijker. Stability analysis of numerical methods for delay differential equations[J]. Numer.Math. 1991, 59: 807-814.
[8] L. Torelli. Stability of numerical methods for delay differential equations[J]. J. Comput. Appl. Math. 1989, 25: 15-26.
[9] T.Koto. A stability property of A-stable natural Runge-Kutta methods for systems of delay differential equations[J]. BIT. 1994, 34: 262-267.
[10] N.Guglielmi. Asymptotic stability barriers for natural Runge-Kutta processes for delay equations[J]. SIMA J.Numer.Anal. 2001, 39: 738-763.
[11] CM. Huang. Numerical analysis of nonlinear delay differential equations[C]. Ph.D. Thesis, China Academy of Engineering Physics, 1999.
[12] C. M. Huang, S. F. Li, H. Y. Hu and G. N. Chen. Nonlinear stability of general linear methods for delay differential equations[J]. BIT. 2002, 42: 380-392.
[13] C.M. Huang, H.Y. Fu.S.F. Li. G.N. Chen. Stability analysis of Runge-Kutta methods for nonlinear delay differential equations[J]. BIT. 1999, 39: 270-280.
[14] C.M. Huang, H.Y. Fu,S.F. Li, G.N. Chen. Stability and error analysis of one-leg methods for nonlinear delay differential equations[J]. J.Comput.Appl.Math.1999, 103: 263-279.
[15] C.J. Zhang, S.Z. Zhou. Nonlinear stability and D-convergence of Runge-Kutta methods for DDEs[J]. J.Comput.Appl.Math. 1997, 85: 225-237.
[16] M. Z. Liu and M. N. Spijker. The stability of the methods in the numerical solution of delay differential equations[J]. IMA J. Numer. Anal. 1990, 10: 31-48.
[17] S.F. Li. B-theory of Runge-Kutta methods for stiff Volterra functional differential equations[J]. SCIENCE IN CHINA (Series A). 2003, 46 (5): 662-674.
[18] S.F. Li. B-theory of general linear methods for stiff Volterra functional differential equations[J]. Appl.Numer. Math. 2005, 53 (1): 57-72.
[19] S.F. Li. Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach space[J]. SCIENCE IN CHINA (Series A).2005, 48 (3): 372-387.
[20] L.P. Wen, Y.X. Yu, S.F. Li. The test problem class of Volterra functional differential equations in Banach space[J]. Appl. Math.Comput. 2006, 179: 30-38.
[21] L.R Wen, Y.X. Yu, S.F. Li. Stability of explicit and diagonal implicit Runge-Kutta methods for nonlinear Volterra functional differential equations in Banach space[J]. Appl. Math.Comput. 2006, 183: 68-78.
[22] L.P. Wen, S.F. Li. Nonlinear stability of linear multistep methods for stiff delay differential equations in Banach space[J]. Appl. Math.Comput. 2005,168: 1031-1044.
[23]李寿佛．Runge-Kutta方法的稳定性准则[J]．湘潭大学自然科学学报．1987， 3：11-17．
[24] K. Burrage and J.C. Butcher. Non-linear stability of a general class of differentialequations methods[J]. BIT. 1980, 20: 185-203.
[25] K. Burrage and J.C. Butcher. Stability criteria for implicit Runge-Kutta methods[J]. SIAM J.Numer.Anal. 1979, 16: 46-57.
[26] J.C. Butcher. Linear and non-linear stability for general linear methods[J]. BIT. 1987, 27: 182-189.
[27] S.F. Li. Nonlinear stability of general linear methods[J]. J.Comput.Math. 1991, 9: 97-104.
[28]肖爱国． Runge-Kutta方法(θ，p，q)-代数稳定性[J].计算数学． 1993，4： 440-448．
[29] M.Zennaro. P-stability of Runge-Kutta methods for delay differential equations[J]. Numer.Math. 1986, 49: 305-318.
[30] M.Zennaro. Asymptotic stability analysis of Runge-Kutta methods for nonlinearsystems of delay differential equations[J]. Numer.Math. 1997, 77: 549-563.
[31] J.K. Hale. Homoclinic orbits and chaos in delay equations[M]. Proceedings of the Ninth Dundee Conference on Ordinary and Partial Differential Equations(B.D. Sleeman and R.J. Jarvis, eds). Wiley, New York, 1986.
[32] Gan Siqing, Zheng Weimin. Stability analysis of Runge-Kutta methods for delay integro-differential equations [J]. Tsinghua Science and Technology. 2004, 2: 185-188.
[33]甘四清．多步Runge-Kutta方法的收敛性与稳定性[J]．长沙铁道学院学报． 1997,3：62-66．
[34]李寿佛．一般线性方法的非线性稳定性[C]．全国首届高校常微分方程数值方 法研讨会．兰州，1987．
[35]肖飞雁，王文强．一类非线性多延迟微分方程一般线性方法的稳定性[J]．西北 师范大学学报(自然科学版)．2004，4：19-22．