Explicit Euler for a scalar neutral delay on a compatible grid

Inputs and admissibility. Take a scalar neutral-delay problem from Scalar neutral delay equations and the derivative-segment shift with $T=K\tau$, $\tau >0$, integers $K,m\ge 1$, a compatible $C^1$ history $\varphi$, coefficient $a$, and right-hand side $f$. An optional budget $B\in {\mathbb{N}}_0$ limits calls to $f$; if omitted, it imposes no limit. These are the admissible inputs.

Output. On completion, return $Y_0,\dots ,Y_N$ and $V_0,\dots ,V_{N-1}$. If the budget expires first, return budget-exhausted, the number of completed updates, the initialized history and all accepted states and derivatives.

1. Set $N=Km$, $h=\tau \mathrm{/}m$, and $t_n=nh$. Store $Y_j=\varphi (t_j)$ and $V_j={\varphi }^{\mathrm{\prime }}(t_j)$ for $j=-m,\dots ,-1$, and set $Y_0=\varphi (0)$.

2. For $n=0,\dots ,N-1$, if a budget was supplied and $n\ge B$, return budget-exhausted with $n$ completed updates, the initialized values, $Y_0,\dots ,Y_n$, and $V_0,\dots ,V_{n-1}$. Otherwise set $D_n={\varphi }^{\mathrm{\prime }}(t_n-\tau )$ when $n<m$ and $D_n=V_{n-m}$ when $n\ge m$, then set $V_n=a{D}_n+f(t_n,Y_n,Y_{n-m})$ and $Y_{n+1}=Y_n+h{V}_n$.

3. Return completed, the nodal states $Y_0,\dots ,Y_N$ and nodal derivative values $V_0,\dots ,V_{N-1}$.

Defined evaluations, compatibility, finite execution and storage counts are proved in Finite execution and storage counts for compatible neutral Euler ↗.