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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) Zbl 1063.65070
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.

##### MSC:
 65L20 Stability and convergence 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 functional-differential equations
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