Stability of Runge-Kutta methods in the numerical solution of equation \(u'(t)=au(t)+a_{0} u([t])+a_{1} u([t-1])\)
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.
scientific article; zbMATH DE number 2136152
scientific article