Explicit Euler with a stored piecewise-linear delay history

Inputs and admissibility. Take a retarded-delay problem from Retarded delay equations and their history space with $\tau >0$, a target $T>0$, a positive integer $m$, its history $\varphi$ and vector field $f$, and an optional evaluation budget $B\in {\mathbb{N}}_0$. The budget counts calls to $f$; if omitted, it imposes no limit. These conditions are the algorithm's admissible inputs.

Output. On completion, return the nodes and their piecewise-linear interpolant through $T_h=Nh$, together with its restriction to $[0,T]$. If the budget expires first, return budget-exhausted, the number of completed updates, all values stored so far, and the piecewise-linear interpolant through the last accepted node.

1. Set $h=\tau \mathrm{/}m$, $N=\lceil T\mathrm{/}h\rceil$ (the least integer with $Nh\ge T$), and $T_h=Nh$, so $0\le T_h-T<h$. Initialize and store $Y_j=\varphi (jh)$ for $j=-m,-m+1,\dots ,0$.

2. For $n=0,1,\dots ,N-1$, if a budget was supplied and $n\ge B$, return budget-exhausted with $n$ completed updates, the stored values $Y_{-m},\dots ,Y_n$, and their interpolant on $[-\tau ,t_n]$. Otherwise evaluate $F_n=f(t_n,Y_n,Y_{n-m})$ and set $Y_{n+1}=Y_n+h{F}_n$. The delayed query is a stored node because $t_n-\tau =t_{n-m}$. Every required evaluation is defined: $n<N$ implies $t_n<T$, and the vector field is defined for every state on the time domain containing $[0,T]$.

3. On each grid cell in $[-\tau ,T_h]$, define ${\widehat{y}}_h$ by linear interpolation between the stored endpoints. Return completed, the nodes $Y_{-m},\dots ,Y_N$, and ${\widehat{y}}_h$ on $[-\tau ,T_h]$ and its restriction to $[0,T]$.

The evaluation counts, history storage, and direct-index interpolation cost are proved in Stored retarded Euler and distributed midpoint work, storage, and grid alignment.