Rational methods for abstract linear initial boundary value problems without order reduction

Given an A A -stable rational approximation to e^z e z of order p p , numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term f f and boundary value g g . These procedures exhibit the optimal order p p and can be implemented by using just one single evaluation of f f and g g per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for p ≤ 6 p ≤ 6 . The full discretization is also studied and the theoretical results are corroborated by numerical experiments.