Strong convergence rate of the implicit midpoint method for stochastic KdV equations
Abstract
We establish a strong convergence rate for the implicit midpoint method applied to the periodic stochastic Korteweg-de Vries equation with additive noise, addressing a question that has remained open since the work of Debussche and Printems. To control nonlinear error growth, we obtain exponential integrability estimates for the exact and comparison solutions using the coercivity of a classical KdV invariant. We then derive local error bounds from estimates for the comparison solutions and control error accumulation through an L2 error propagation identity that isolates a martingale increment. Under H09/2 regularity assumptions on the noise and the deterministic initial data, we prove that, on every fixed finite time interval, the maximum L2(𝕋) error at the time points is O(h) in Lp(Ω) for every 1≤ p<∞. Notably, the exponential integrability and the convergence rate hold for every damping coefficient λ≥0, including the undamped case λ=0. Numerical experiments confirm the predicted first-order convergence.
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BibTeX
@misc{li2026strong,
title = {{Strong convergence rate of the implicit midpoint method for stochastic KdV equations}},
author = {Junjie Li and Xinyan Niu and Chun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17600},
eprint = {2610.17600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17600},
}
RIS
TY - UNPB
TI - Strong convergence rate of the implicit midpoint method for stochastic KdV equations
AU - Junjie Li
AU - Xinyan Niu
AU - Chun Li
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17600
UR - https://arxiv.org/abs/2610.17600
AB - We establish a strong convergence rate for the implicit midpoint method applied to the periodic stochastic Korteweg-de Vries equation with additive noise, addressing a question that has remained open since the work of Debussche and Printems. To control nonlinear error growth, we obtain exponential integrability estimates for the exact and comparison solutions using the coercivity of a classical KdV invariant. We then derive local error bounds from estimates for the comparison solutions and control error accumulation through an L^2 error propagation identity that isolates a martingale increment. Under H_0^9/2 regularity assumptions on the noise and the deterministic initial data, we prove that, on every fixed finite time interval, the maximum L^2(𝕋) error at the time points is O(h) in L^p(Ω) for every 1≤ p<∞. Notably, the exponential integrability and the convergence rate hold for every damping coefficient λ≥0, including the undamped case λ=0. Numerical experiments confirm the predicted first-order convergence.
ER -