Asymptotic expansions for the midpoint rule applied to delay differential equations

Let $u^ * $ be the solution of the delay differential equation \[ \begin{gathered} u'(t) = f(t,u(t),u(t - \tau ))\quad {\text{for}}\,t > 0, \hfill \\ u(t) = \varphi (t)\qquad{\text{for}} - \tau \leqq t \leqq 0. \hfill \\ \end{gathered} \]For $t_j = jh$, let $u_h (t_j )$ be the approximation for $u^ * (t_j )$ obtained by the explicit midpoint rule, initialized by one Euler step. It is shown that if f and $\varphi $ are sufficiently differentiable and ${\tau / {h \in \mathbb{N}}}$, there are functions $e_k $ such that for j even, \[u_h \left( {t_j } \right) - u^ * \left( {t_j } \right) = \sum _{k = 1}^r {h^{2k} e_k } \left( {t_j } \right) + O\left( {h^{2r + 2} } \right).\] This expansion of the global discretization error exists in spite of the fact that the solution $u^ * $ usually has jump discontinuities in its derivatives at the points $t = n\tau $, $n \in \mathbb{N}$. Thus repeated extrapolation to the limit (Richardson extrapolation) may be applied to improve the approximation. Furthermore, it is show...