A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations

This paper deals with adapting Runge-Kutta methods to differential equations with a lagging argument. A new interpolation procedure is introduced which leads to numerical processes that satisfy an important asymptotic stability condition related to the class of testproblemsU(t)=U(t)+U(t–) with , C, Re()<–||, and >0. Ifc_i denotes theith abscissa of a given Runge-Kutta method, then in thenth stept_n–1t_n:=t_n–1+h of the numerical process our interpolation procedure computes an approximation toU(t_n–1+c_ih–) from approximations that have already been generated by the process at pointst_j–1+c_ih(j=1,2,3,...). For two of these new processes and a standard process we shall consider the convergence behaviour in an actual application to a given, stiff problem.