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

Abstract Given an $$\varvec{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> -stable rational approximation to $$\varvec{e}^{\varvec{z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> </mml:msup> </mml:math> of order $$\varvec{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> , numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term $$\varvec{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> and boundary value $$\varvec{g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> . These procedures exhibit the optimal order $$\varvec{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> and can be implemented by using just one single evaluation of $$\varvec{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> and $$\varvec{g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for $$\varvec{p \le 6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> . The full discretization is also studied and the theoretical results are corroborated by numerical experiments.