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Long-time stability and convergence of a BDF3 method for a parabolic equation with fading memory

  • Wenlin Qiu
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Abstract

We analyze the long-time stability and convergence of a third-order backward differentiation formula (BDF3) for parabolic equations with exponentially fading memory. The BDF3 scalar resolvent has oscillatory coefficients, so the monotonicity argument used for lower-order formulas in [Math. Comp. 62 (1994) 477-496] does not apply. We instead control memory feedback by the absolute mass of its discrete Green sequence and establish the rigorous stability-constant bounds 23/20<Λ3<579/500. A causal third-order memory quadrature with nonnegative assembled weights then yields exponentially weighted stability without a parabolic step-size restriction. With third-order starting values, we prove a uniform-in-time third-order temporal error estimate and extend it to conforming finite elements. Numerical tests confirm the predicted convergence orders and the long-time behavior.

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Cite

BibTeXRIS
BibTeX
@misc{qiu2026longtime,
  title = {{Long-time stability and convergence of a BDF3 method for a parabolic equation with fading memory}},
  author = {Wenlin Qiu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32364},
  eprint = {2610.32364},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32364},
}
RIS
TY  - UNPB
TI  - Long-time stability and convergence of a BDF3 method for a parabolic equation with fading memory
AU  - Wenlin Qiu
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32364
UR  - https://arxiv.org/abs/2610.32364
AB  - We analyze the long-time stability and convergence of a third-order backward differentiation formula (BDF3) for parabolic equations with exponentially fading memory. The BDF3 scalar resolvent has oscillatory coefficients, so the monotonicity argument used for lower-order formulas in [Math. Comp. 62 (1994) 477-496] does not apply. We instead control memory feedback by the absolute mass of its discrete Green sequence and establish the rigorous stability-constant bounds 23/20<Λ_3<579/500. A causal third-order memory quadrature with nonnegative assembled weights then yields exponentially weighted stability without a parabolic step-size restriction. With third-order starting values, we prove a uniform-in-time third-order temporal error estimate and extend it to conforming finite elements. Numerical tests confirm the predicted convergence orders and the long-time behavior.
ER  -