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Zbl 1063.65070
Yang, Zhanwen; Liu, Mingzhu; Song, Minghui
Stability of Runge-Kutta methods in the numerical solution of equation $u'(t)=au(t)+a_{0} u([t])+a_{1} u([t-1])$. (English)
[J] Appl. Math. Comput. 162, No. 1, 37-50 (2005). ISSN 0096-3003

The authors discuss the numerical solution of the initial value problem $u'(t) = a u(t) + a_0 u([t]) + a_1 u([t-1])$, $u(0) = u_0$, $u(-1) = u_{-1}$, where $[\cdot]$ denotes the floor function (round down to nearest integer). This is a special case of a delay differential equation with piecewise continuous argument. The numerical methods under consideration are of Runge-Kutta type. The authors first explain how standard Runge-Kutta methods can be applied to this class of problems. Then, the asymptotic stability of various special types of Runge-Kutta methods (e.g., Gauss, Lobatto, and Radau) is investigated.
[Kai Diethelm (Braunschweig)]

MSC 2000:: *65L20 Stability of numerical methods for ODE
65L06 Multistep, Runge-Kutta, and extrapolation methods
65L10 Boundary value problems for ODE (numerical methods)
34K28 Numerical approximation of solutions of FDE

Keywords: delay differential equation; piecewise continuous argument; Runge-Kutta method; asymptotic stability

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