On the asymptotics of the trapezoidal rule for the pantograph equation

The paper deals with the trapezoidal rule discretization of a class of linear delay differential equations, with a special emphasis on equations with a proportional delay. Our purpose is to analyse the asymptotic properties of the numerical solutions and formulate their upper bounds. We also survey the known results and show that our formulae improve and generalize these results. In particular, we set up conditions under which the numerical solution of the scalar pantograph equation has the same decay rate as the exact solution.