Stability of Runge-Kutta methods in the numerical solution of linear impulsive differential equations. (English) Zbl 1193.65121

Summary: This paper deals with the stability analysis of the analytic and numerical solutions of linear impulsive differential equations. The numerical method with variable stepsize is defined, the conditions that the numerical solutions preserve the stability property of the analytic ones are obtained and some numerical experiments are given.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A37 Ordinary differential equations with impulses


Full Text: DOI


[1] Akhmet, M.U., On the general problem of stability for impulsive differential equations, J. math. anal. appl., 288, 182-196, (2003) · Zbl 1047.34094
[2] Bainov, D.D.; Simeonov, P.S., Systems with impulsive effect: stability, theory and applications, (1989), Ellis Horwood Chichester · Zbl 0676.34035
[3] Butcher, J.C., The numerical analysis of ordinary differential equations: runge – kutta and general linear methods, (1987), Wiley New York · Zbl 0616.65072
[4] Dekker, K.; Verwer, J.G., Stability of runge – kutta methods for stiff nonlinear differential equations, (1984), North-Holland Amsterdam · Zbl 0571.65057
[5] Dong, L.Z.; Chen, L.; Sun, L.H., Extiction and permanence of the predator-prey system with stocking of prey and harvesting of predator impulsively, Math. meth. appl. sci., 29, 415-425, (2006) · Zbl 1086.92051
[6] Hairer, E.; Wanner, G., Solving ordinary differential equations II, Stiff and differential algebraic problems, (1996), Springer New York · Zbl 0859.65067
[7] Iserles, A.; Nørsett, S.P., Order stars and rational approximations to exp(z), Appl. numer. math., 5, 63-70, (1989) · Zbl 0674.65043
[8] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[9] Lambert, J.D., Numerical methods for ordinary differential systems, (1991), Wiley New York · Zbl 0745.65049
[10] M.Z. Liu, S.F. Ma, Z.W. Yang, Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.12.008. · Zbl 1193.65122
[11] Mil’man, V.D.; Myshkis, A.D., On the stability of motion in the presence of impulses, Sib. math. J., 1, 233-237, (1960) · Zbl 1358.34022
[12] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[13] Wanner, G.; Hairer, E.; Nørsett, S.P., Order stars and stability theorems, Bit, 18, 475-489, (1978) · Zbl 0444.65039
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