Strong convergence rate of the implicit midpoint method for stochastic KdV equations
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 $L^2(\mathbb{T})$ error at the time points is $O(h)$ in $L^p(Ω)$ for every $1\le p<\infty$. Notably, the exponential integrability and the convergence rate hold for every damping coefficient $λ\ge0$, including the undamped case $λ=0$. Numerical experiments confirm the predicted first-order convergence.