One-step ODE methods, local defect, uniform order, and step stability

Let $d\ge 1$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$ be open, and consider a first-order IVP and solution in the sense of First-order systems, initial value problems, and solutions on intervals. A one-step method is a family of maps ${\mathrm{\Psi }}_h(t,v)$, defined for admitted times $t$, positive step sizes $h$, and states $v$, whose value is the method's approximation at time $t+h$ from the single input state $v$ at time $t$. It does not use earlier numerical states.

If $y$ is an exact solution on an interval containing $[t,t+h]$, the local defect is

${\delta }_h(t;y):={\mathrm{\Psi }}_h(t,y(t))-y(t+h)\mathrm{.}$

Let $K\subseteq D$ be a fixed compact time-state set. The method has uniform local order $p$ on $K$ along $y$ if there are constants $C\ge 0$ and $h_0>0$, independent of the admitted starting time $t$ and step $0<h\le h_0$, such that

$\mathrm{\parallel }{\delta }_h(t;y){\mathrm{\parallel }}_2\le C{h}^{p+1}$

whenever the exact solution graph on $[t,t+h]$ and the associated method time-state inputs, stages, and outputs under consideration lie in $K$. Here $p$ is a positive integer and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ is the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

The method has one-step Lipschitz stability on $K$ with constant $L\ge 0$ if

$\mathrm{\parallel }{\mathrm{\Psi }}_h(t,u)-{\mathrm{\Psi }}_h(t,v){\mathrm{\parallel }}_2\le (1+Lh)\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$

for every admitted step size $h$, admitted time $t$, and input states $u,v$ with $(t,u),(t,v)\in K$. This is a bound on the change in one step; it is separate from absolute or linear stability regions and does not itself assert a global error estimate.