Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays

Abstract

This paper is concerned with the study of the stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays. We are interested in the comparison between the analytical and numerical stability regions. First, we focus on scalar equations with real coefficients. It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical solution. Then, we consider the multidimensional case. A new stability condition for the stability of the analytical solution is given. Under this condition, the asymptotic stability of Gauss-Pouzet methods is investigated.

Keywords

Volterra delay integro-differential equation / asymptotic stability / Runge-Kutta-Pouzet method

References

[1] Baker C T H, Ford N J. Some applications of the boundary-locus method and the method of D-partitions. IMA J Numer Anal, 1991, 11: 143-158

[2] Baker C T H, Ford N J. Stability properties of a scheme for the approximate solution of a delay-integro-differential equations. Appl Numer Math, 1992, 9: 357-370

[3] Bellen A, Zennaro M. Numerical Methods for Delay Differential Equations. Oxford: Oxford University Press, 2003

[4] Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge Monographs on Applied and Computational Mathematics, Vol 15. Cambridge: Cambridge University Press, 2004

[5] Brunner H, Hairer E, Norsett S P. Runge-Kutta theory for Volterra integral equations of the second kind. Math Comp, 1982, 39: 147-163

[6] Brunner H, Lambert J D. Stability of numerical methods for Volterra integro-differential equations. Computing, 1974, 12: 75-89

[7] Calvo M, Grande T. On the asymptotic stability of Θ-methods for delay differential equations. Numer Math, 1988, 54: 257-269

[8] Cryer C W. Highly stable multistep methods for retarded differential equations. SIAM J Numer Anal, 1974, 11: 788-797

[9] Diekmann O, Van Gils S A, Verduin Lunel S M, Walther H-O. Delay equations: Functional-, Complex-, and Nonlinear Analysis. Berlin: Springer-Verlag, 1995

[10] Guglielmi N. On the asymptotic stability properties of Runge-Kutta methods for delay differential equations. Numer Math, 1997, 77: 467-485

[11] Guglielmi N. Delay dependent stability regions of Θ-methods for delay differential equations. IMA J Numer Anal, 1998, 18: 399-418

[12] Guglielmi N. Asymptotic stability barriers for natural Runge-Kutta processes for delay equations. SIAM J Numer Anal, 2001, 39: 763-783

[13] Guglielmi N, Hairer E. Order stars and stability for delay differential equations. Numer Math, 1999, 83: 371-383

[14] Guglielmi N, Hairer E. Geometric proofs of numerical stability for delay equations. IMA J Numer Anal, 2001, 21: 439-450

[15] Hairer E, Wanner G. Solving Ordinary Differential equations II. Berlin: Springer, 1991

[16] Hale J K, Verduyn Lunel S M. Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993

[17] Hu G, Mitsui T. Stability analysis of numerical methods for systems of neutral delay-differential equations. BIT, 1995, 35: 504-515

[18] Huang C. Stability of linear multistep methods for delay integro-differential equations. Comput Math Appl, 2008, 55: 2830-2838

[19] Huang C, Vandewalle S. An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J Sci Comput, 2004, 25: 1608-1632

[20] in’t Hout K J. Stability analysis of Runge-Kutta methods for systems of delay differential equations. IMA J Numer Anal, 1997, 17: 17-27

[21] Koto T. A stability property of A-stable natural Runge-Kutta methods for systems of delay differential equations. BIT, 1994, 34: 262-267

[22] Koto T. Stability of Runge-Kutta methods for delay integro-differential equations. J Comput Appl Math, 2002, 145: 483-492

[23] Li S. Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces. Sci China, Ser A, 2005, 48: 372-387

[24] Lubich Ch. Runge-Kutta theory for Volterra integrodifferential equations. Numer Math, 1982, 40: 119-135

[25] Luzyanina T, Engelborghs K, Roose D. Computing stability of differential equations with bounded distributed delays. Numer Algorithms, 2003, 34: 41-66

[26] Maset S. Stability of Runge-Kutta methods for linear delay differential equations. Numer Math, 2000, 87: 355-371

[27] Maset S. Instability of Runge-Kutta methods when applied to linear systems delay differential equations. Numer Math, 2002, 90: 555-562

[28] Matthys J. A-stable linear multistep methods for Volterra integro-differential equations. Numer Math, 1976, 27: 85-94

[29] Sidibe B S, Liu M Z. Asymptotic stability for Gauss methods for neutral delay differential equation. J Comput Math, 2002, 20: 217-224

[30] Wu S, Gan S. Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput Math Appl, 2008, 55: 2426-2443

[31] Zhang C, Vandewalle S. Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations. Adv Comput Math, 2008, 28: 383-399

[32] Zhao J J, Xu Y, Liu M Z. Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system. Appl Math Comput, 2005, 167: 1062-1079

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