Optimal convergence of a second order low-regularity integrator for the KdV equation

Abstract: In this paper, we establish the optimal convergence result of a second order exponential-type integrator from (136, Numer. Math., 2017) for solving the KdV equation under rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides the second order accuracy in $H^\gamma$ for initial data in $H^{\gamma+4}$ for any $\gamma\geq0$, where the regularity requirement is lower than the classical methods. The result is confirmed by numerical experiments and comparisons are made with the Strang splitting scheme.