Composite midpoint quadrature for a distributed delay history

Inputs and admissibility. Take a delay $\tau >0$, an integer $m\ge 1$, a continuous kernel $K:[0,\tau ]\to \mathbb{R}$, and the completed stored piecewise-linear history ${\widehat{y}}_h:[-\tau ,T_h]\to {\mathbb{R}}^d$ from Explicit Euler with a stored piecewise-linear delay history, with $h=\tau \mathrm{/}m$ and $T_h=Nh$ for an integer $N\ge 1$. Choose an integer refinement factor $q\ge 1$ and a query index $n\in \{0,\dots ,N\}$. Then $t_n=nh$ and $t_n-\theta \in [-\tau ,T_h]$ for every $\theta \in [0,\tau ]$, so all evaluations below are defined.

Output. Return the vector ${\widehat{\mathcal{I}}}_{K,h,r}(t_n)\in {\mathbb{R}}^d$ given by the following finite computation.

1. Set $r=qm$, $\mathrm{\Delta }=\tau \mathrm{/}r=h\mathrm{/}q$, initialize $A=0\in {\mathbb{R}}^d$, and set ${\theta }_j=(j+1\mathrm{/}2)\mathrm{\Delta }$ for each $j=0,\dots ,r-1$.

2. For $j=0,\dots ,r-1$, add $\mathrm{\Delta }K({\theta }_j){\widehat{y}}_h(t_n-{\theta }_j)$ to $A$.

3. Return $A$, equivalently ${\widehat{\mathcal{I}}}_{K,h,r}(t_n)=\mathrm{\Delta }{\sum }_{j=0}^{r-1}K({\theta }_j){\widehat{y}}_h(t_n-{\theta }_j)\mathrm{.}$

Each history query uses the piecewise-linear interpolant, including queries between Euler nodes.

The evaluation, arithmetic and working-storage counts, and the alignment of Euler-history knots with quadrature-cell endpoints, are proved in Stored retarded Euler and distributed midpoint work, storage, and grid alignment.