Rational methods for abstract linear, non-homogeneous problems without order reduction

Starting from an A-stable rational approximation to ${\mathrm{e}}^z$ of order $p$, $$r(z)=1+z+\cdots +z^p\mathrm{/}p!+O(z^{p+1}),$$ families of stable methods are proposed to time discretize abstract IVP’s of the type $u^{\mathrm{\prime }}(t)=Au(t)+f(t)$. These numerical procedures turn out to be of order $p$, thus overcoming the order reduction phenomenon, and only one evaluation of $f$ per step is required.