Numerical solution of retarded functional differential equations as abstract Cauchy problems. (Q1421198)

In this paper an extension of the numerical methods given by \textit{A. Bellen} and \textit{S. Maset} [Num. Math. 84, 351--374 (2000; Zbl 0949.65072)] for the numerical solution of delay differential equations (DDEs) is proposed. Given an initial value problem for a linear system of DDEs with a constant delay \( \tau\): \(y'(t)= L y(t) + M y(t- \tau), t \geq 0,\) \( y(t) = \varphi (t), t \in [- \tau, 0],\) if \( y \) is continuously differentiable, this problem can be reformulated as an abstract Cauchy problem defined by the hyperbolic partial differential equation for \( V(t, \theta)= y(t + \theta)\): \( V_t = V_{\theta}, t \geq 0, \theta \in [ - \tau,0]\) with \( V_{\theta} (t, \theta)= L V(t,0)+ M V(t, - \tau), t \geq 0\) and \( V(0, \theta) = \varphi ( \theta), \theta \in [- \tau, 0]\). Then by introducing a mesh in \( [ - \tau, 0]\) and a discretization in the \( \theta\)-variable, the author proposes some families of implicit Runge-Kutta methods to advance the numerical solution in the \(t\)-variable. This approach is extended to non linear DDEs and a complete analysis of the convergence of a class of implicit Runge-Kutta methods which includes Radau IIA family is carried out. Further the results of some numerical experiments are given to confirm the theory for problems with a sufficiently smooth solution. The paper includes a section on the preservation of the asymptotic stability of Runge-Kutta methods for DDES. It is found that under some restrictions on the step size the class of Runge-Kutta methods under consideration preserves the stability of the original problem.