High-Accuracy Stable Difference Schemes for Well-Posed Initial-Value Problems

For $t > 0$, let $u(t)$ satisfy \[ \frac{{du}}{{dt}} = Au, \] where A is a linear operator, the generator of a strongly continuous semigroup. Let $v(t)$ satisfy $v(t + h) = r(hA)v(t)$, where $r( \cdot )$ is a rational function of one variable, such that $| {r(z)} | \leqq 1$ for all $\operatorname{Re} z \leqq 0$. Let $u(0) = v(0)$ and be in the domain of $A^{q + 2} $. Then, if \[ \left| {r(z) - e^z } \right| = O\left( {| z |^{q + 1} } \right)\quad {\text{as }} | z | \to 0,\] we have \[ \| {u(t) = v(t)} \| = O\left( {h^q } \right)\] as $h \to 0$, where q is an arbitrary nonnegative integer.