Arbitrary-Order Pad\'e-Closed Anchored Two-Derivative Time Discretizations: $s$ Active Stages, Order $2s$, and $L$-Stability

Abstract

An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Pad\'{e} approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Pad\'{e}--Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss--Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.