Implicit residuals and their update and derivative units

Let $G:U\to {\mathbb{R}}^m$ be the residual map of an implicit equation, where $U\subseteq {\mathbb{R}}^m$ is the admissible set for the unknown vector. An exact solution is a point $z^{\ast }\in U$ with $G(z^{\ast })=0$.

An equation residual need not have the same units as its unknown. When the equation is normalized so that each component of $G(z)$ has the units of the corresponding component of $z$, the update residual at an approximate solution $\tilde{z}\in U$ is $G(\tilde{z})$. For example, $z^2-a$ has squared state units when $a$ does, so it must first be normalized to use this update-residual convention. If the original equation was divided by a positive step scale $\sigma$, then the derivative residual is $G(\tilde{z})\mathrm{/}\sigma$, and the corresponding update residual is $\sigma$ times that derivative residual. A residual norm is not a forward-error bound unless a separate inverse or stability estimate converts it to $\mathrm{\parallel }\tilde{z}-z^{\ast }\mathrm{\parallel }$.

For a first-order vector field $f:D\to {\mathbb{R}}^d$ on an open time-state domain $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$ (First-order systems, initial value problems, and solutions on intervals), take a time $t$, input $y\in {\mathbb{R}}^d$, and step $h>0$ with $(t,y)\in D$. For backward Euler, an endpoint $z$ is admissible only when $(t+h,z)\in D$, and its update residual is

$G_{\mathrm{B}\mathrm{E}}(z)=z-y-hf(t+h,z),$

so the derivative residual is $G_{\mathrm{B}\mathrm{E}}(z)\mathrm{/}h$.

For an $s$-stage implicit Runge–Kutta tableau $(A,b,c)$, a stage vector $Y=(Y_1,\dots ,Y_s)$ is admissible only when $(t+c_{j}h,Y_j)\in D$ for every $j$. On that admissible set, its stage update residual has blocks

$G_i(Y)=Y_i-y-h{\sum }_{j=1}^s{a}_{ij}f(t+c_{j}h,Y_j),i=1,\dots ,s\mathrm{.}$

The stage derivative residual is $G(Y)\mathrm{/}h$. These conventions distinguish an equation residual from the distance to its exact root.