Slow--Fast Response Correction for a Compact Fourth-Order IMEX Two-Derivative Method

@misc{huo2026slowfast,
  title = {{Slow--Fast Response Correction for a Compact Fourth-Order IMEX Two-Derivative Method}},
  author = {Zhixin Huo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27761},
  eprint = {2609.27761},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27761},
}

TY  - UNPB
TI  - Slow--Fast Response Correction for a Compact Fourth-Order IMEX Two-Derivative Method
AU  - Zhixin Huo
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.27761
UR  - https://arxiv.org/abs/2609.27761
AB  - A compact two-stage fourth-order two-derivative IMEX method may retain its classical non-commuting fourth-order expansion and L-stable stiff damping while losing high-order accuracy when the relaxation time is comparable with the time step. We ask whether uniform fourth-order accuracy can be recovered without redesigning the original two-stage, two-implicit-solve core. For linear relaxation systems with a separated slow spectral subspace, we introduce a slow--fast response correction (SFRC): the exact slow projection is advanced by the unchanged compact IMEX step and projected back to the slow subspace, while the fast complement is advanced by its exact semigroup. For fixed nonzero relaxation time, the correction is O(dt^5), so the base mixed expansion through degree four is unchanged, and the pure-fast response remains L-stable. An abstract slow--fast argument reduces the full-state error to a slow-block defect and converts a uniform O(dt^5) local estimate into a uniform O(dt^4) global estimate. For each Jin--Xin Fourier mode, the required local estimate is proved by an exact projected-amplification factorization and a uniform bound in x=eps/dt. In particular, the unchanged compact core already has a uniform fifth-order one-step defect after restriction to the exact finite-eps slow eigenspace. A two-dimensional three-variable model yields the same mechanism on fixed Fourier grids. Symbolic verification and one- and two-dimensional experiments corroborate the analytical predictions. SFRC is therefore a rigorous linear benchmark identifying the finite-eps slow--fast information sufficient to reconcile compactness, mixed fourth-order accuracy, stiff damping, and uniform fourth-order time accuracy.
ER  -