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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)=au(t)+a 0 u([t])+a 1 u([t-1]), u(0)=u 0 , u(-1)=u -1 , where [·] 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.
65L20Stability and convergence of numerical methods for ODE
65L06Multistep, Runge-Kutta, and extrapolation methods
65L10Boundary value problems for ODE (numerical methods)
34K28Numerical approximation of solutions of functional-differential equations