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Linear stability of general linear methods for systems of neutral delay differential equations. (English) Zbl 0989.65078

The authors consider the numerical solution of delay differential equations (DDEs). They investigate linear stability properties of a more general linear method with a type of interpolation procedure for systems of neutral DDEs with multiple delays, and unify many extant results.

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K40 Neutral functional-differential equations
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References:

[1] Barwell, V. K., Special stability problems for functional equations, BIT, 15, 130-135 (1975) · Zbl 0306.65044
[2] Bellen, A.; Jackiewicz, Z.; Zennaro, M., Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52, 605-619 (1988) · Zbl 0644.65049
[3] Bickart, T. A., \(P\)-stable and \(P\)[α, β]-stable integration/interpolation methods in the solution of retarded differential-difference equations, BIT, 22, 464-476 (1982) · Zbl 0531.65044
[4] Da Hu, G.; Mitsui, T., Stability analysis of numerical methods for systems of neutral delay-differential equations, BIT, 35, 504-515 (1995) · Zbl 0841.65062
[5] in’t Hout, K. J., A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT, 32, 634-649 (1992) · Zbl 0765.65069
[6] in’t Hout, K. J., Stability analysis of Runge-Kutta methods for systems of delay differential equations, IMA J. Numer. Anal., 17, 17-27 (1997) · Zbl 0867.65046
[7] in’t Hout, K. J.; Spijker, M. N., Stability analysis of numerical methods for delay differential equations, Numer. Math., 59, 807-814 (1991) · Zbl 0724.65084
[8] Koto, T., A stability property of \(A\)-stable natural Runge-Kutta methods for systems of delay differential equations, BIT, 34, 262-267 (1994) · Zbl 0805.65083
[9] Kuang, J.; Xiang, J.; Tain, H., The asymptotic stability of one-parameter methods for neutral differential equations, BIT, 34, 400-408 (1994) · Zbl 0814.65078
[10] Liu, M. Z.; Spijker, M. N., The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. Numer. Anal., 10, 31-48 (1990) · Zbl 0693.65056
[11] Watanabe, D. S.; Roth, M. G., The stability of difference formulas for delay differential equations, SIAM J. Numer. Anal., 22, 132-145 (1985) · Zbl 0571.65075
[12] Zennaro, M., On the \(P\)-stability of one-step collocation for delay differential equations, ISNM, 74, 334-343 (1985)
[13] Zennaro, M., \(P\)-stability of Runge-Kutta methods for delay differential equations, Numer. Math., 49, 305-318 (1986) · Zbl 0598.65056
[14] Da Hu, G.; Di Hu, G.; Meguid, S. A., Stability of Runge-Kutta methods for delay differential systems with multiple delays, IMA J. Numer. Anal., 19, 349-356 (1999) · Zbl 0948.65079
[15] Qiu, L.; Yang, B.; Kuang, J., The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations, Numer. Math., 81, 451-459 (1999) · Zbl 0918.65061
[16] Zhang, C. J.; Zhou, S. Z., The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations, Science in China, 41, 11, 504-515 (1998)
[17] Zhang, C. J.; Zhou, S. Z., Stability analysis of LMMs for systems of neutral multidelay-differential equations, Computers Math. Applic., 38, 3/4, 113-117 (1999) · Zbl 0940.65085
[18] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations (1987), John Wiley: John Wiley New York · Zbl 0616.65072
[19] Hairer, E.; Norsett, S. P.; Wanner, G., (Solving Ordinary Differential Equations I (1993), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0789.65048
[20] Hairer, E.; Wanner, G., (Solving Ordinary Differential Equations II (1996), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0859.65067
[21] Li, S., Theory of Computational Methods for Stiff Differential Equations (1997), Hunan Science and Technology: Hunan Science and Technology Changsha
[22] Huang, C.; Li, S.; Fu, H.; Chen, G., Stability and error analysis of one-leg methods for nonlinear delay differential equations, J. Comput. Appl. Math., 103, 263-279 (1999) · Zbl 0948.65078
[23] Stetter, H. J., Numerische Losung von Differentialgleichungen mit nacheilendem Argument, ZAMM, 45, 79-80 (1965) · Zbl 0208.41703
[24] Oberle, H. J.; Pesh, H. J., Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math., 37, 235-255 (1981) · Zbl 0469.65057
[25] Zennaro, M., Natural continuous extensions of Runge-Kutta methods, Math. Comp., 46, 119-133 (1986) · Zbl 0608.65043
[26] Strang, G., Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys., 41, 147-154 (1962) · Zbl 0111.31601
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