A parity anomaly in higher-order diagonal Padé time-stepping schemes

We extend the Padé time-stepping (PTS) framework of Amundsen and Bruno through the underlying Toeplitz structure to the higher-order diagonal schemes PTS[n/n] for n=3,4,5,6, of formal order 2n. On the oscillatory family A2(θ,κ)=V(θ)B2(κ)V(θ)T the componentwise map is homogeneous of degree one, reducing stability to a polar iteration; a Padé rescaling argument proves the shrinkage h∗(κ)∼Cn/κ for all n≥1. The boundary geometry exhibits a parity anomaly: even n yield smooth near-circular contours, odd n singular boundaries with narrow inward excursions on a near-circular base. The mechanism is the identity A22k=(-κ)kI, which forces a parity split in the entries of the Toeplitz matrix Tn and gives the singularity dichotomy that Tn is singular iff n is odd and Qm=0; a Padé-of-cosine identity frames the phenomenon as a classical limitation of diagonal Padé approximation. The anomaly persists on a discretized variable-coefficient acoustic wave equation, where the stable-step ordering hPTS[3/3]∗<hPTS[5/5]∗<hPTS[4/4]∗<hPTS[6/6]∗ survives cost normalization, with the order-10 odd scheme more restrictive than the order-8 even one, and is already present at constant coefficients, identifying the anomaly as structural. On a nonlinear reaction–diffusion benchmark, PTS[n/n] reaches target accuracies at roughly an order of magnitude lower wall-clock cost than matched-order multiderivative Hermite IMEX schemes.