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

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