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Stability of multistep Runge-Kutta methods for systems of functional-differential and functional equations. (English) Zbl 1068.65105

The aim of the paper is the stability of multistep Runge-Kutta methods applied to systems of linear functional-differential and functional equations.
The first part concerns the problem statement and the principal results existing in the literature concerning the stability of difference formulas for the systems of functional-differential and functional equations.
The second part is devoted to the numerical methods, more precisely the multistep Runge-Kutta methods. One presents the numerical process obtained by applying one of these Runge-Kutta methods to the considered system of functional-differential and functional equations presented in the first section.
The last part is devoted to the stability analysis of the above numerical process. Some results concerning the relation between the stability of the numerical process and the \(A\)-stability of the numerical Runge-Kutta method which generates this one are presented.

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L05 Numerical methods for initial value problems involving ordinary differential equations
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References:

[1] Huang, C. M.; Chang, Q. S., Stability analysis of numerical methods for systems of functional-differential and functional equations, Computers Math. Applic., 44, 5/6, 717-729 (2002) · Zbl 1035.65089
[2] Liu, Y., Runge-Kutta-collocation methods for system of functional-differential and functional equations, Adv. Comput. Math., 11, 315-329 (1999) · Zbl 0948.65069
[3] Burrage, K., High order algebraically stable multistep Runge-Kutta methods, SIAM J. Numer. Anal., 24, 106-115 (1987) · Zbl 0611.65046
[4] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0859.65067
[5] Li, S. F., Stability and B-convergence properties of multistep Runge-Kutta methods, Math. Comp., 69, 1481-1504 (1999) · Zbl 0954.65067
[6] Iserles, A.; Strang, G., The optimal accuracy of difference schemes, Trans. Amer. Math. Soc., 277, 779-803 (1983) · Zbl 0573.65071
[7] Strang, G., Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys., 41, 147-154 (1962) · Zbl 0111.31601
[8] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0729.15001
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