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

Starting from an A-stable rational approximation to eᶻ of order p, r(z)= 1+ z+ ⋯ + zᵖ/ p! + O(zᵖ⁺¹), families of stable methods are proposed to time discretize abstract IVP's of the type $u'(t) = A u(t) + f(t)$. 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.