High order stable Runge–Kutta methods for nonlinear generalized pantograph equations on the geometric mesh

This paper deals with the Runge–Kutta methods discretization of a class of nonlinear neutral delay differential equations, with a special emphasis on equations with a proportional delay. In order to solve the storage problem and avoid the interpolation procedure we use a geometric mesh (fully-geometric mesh or quasi-geometric mesh). Our purpose is to analyse the stability properties of the numerical solution, and we will identify conditions which imply that the solution is boundedly stable or that it is asymptotically stable. We also give some numerical examples which confirm our results.