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PreprintarXiv licencearXiv

Stability and Error Estimates with Optimal ε-Dependence for Strang Splitting of the Allen--Cahn Equation

  • Rongfu Luo
  • Chaoyu Quan
  • Zhijun Tan
  • Yanyao Wu
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Abstract

In this work, long-time Sobolev stability and error estimates with optimal ε-dependence are established for the Strang splitting method applied to the Allen--Cahn equation with periodic boundary conditions. By combining the modified energy dissipation law, and Hölder-Gagliardo-Nirenberg product estimates, the numerical solution is proven to admit uniform-in-time Hr-seminorm bounds 𝒪(ε−r+1), consistent with the corresponding Sobolev bounds of the exact solution. A second-order error estimate 𝒪(ε2τ2) is further established on any prescribed finite time interval, exhibiting quadratic dependence on ε. Numerical experiments are presented to verify the optimal ε-dependence of both the stability and error estimates.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026stability,
  title = {{Stability and Error Estimates with Optimal $\varepsilon$-Dependence for Strang Splitting of the Allen--Cahn Equation}},
  author = {Rongfu Luo and Chaoyu Quan and Zhijun Tan and Yanyao Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14867},
  eprint = {2610.14867},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14867},
}
RIS
TY  - UNPB
TI  - Stability and Error Estimates with Optimal ε-Dependence for Strang Splitting of the Allen--Cahn Equation
AU  - Rongfu Luo
AU  - Chaoyu Quan
AU  - Zhijun Tan
AU  - Yanyao Wu
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14867
UR  - https://arxiv.org/abs/2610.14867
AB  - In this work, long-time Sobolev stability and error estimates with optimal ε-dependence are established for the Strang splitting method applied to the Allen--Cahn equation with periodic boundary conditions. By combining the modified energy dissipation law, and Hölder-Gagliardo-Nirenberg product estimates, the numerical solution is proven to admit uniform-in-time H^r-seminorm bounds 𝒪(ε^−r+1), consistent with the corresponding Sobolev bounds of the exact solution. A second-order error estimate 𝒪(ε^2τ^2) is further established on any prescribed finite time interval, exhibiting quadratic dependence on ε. Numerical experiments are presented to verify the optimal ε-dependence of both the stability and error estimates.
ER  -