An unfiltered low-regularity integrator for the KdV equation with solutions below ${\bf H^1}$

This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$. New harmonic analysis tools, including new averaging approximations to the exponential phase functions, new frequency decomposition techniques, and new trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new techniques are introduced to establish stability estimates of time discretizations under low-regularity conditions without using filters when the energy techniques fail. The proposed method is proved to be convergent with order $\gamma$ (up to a logarithmic factor) in $L^2$ under the regularity condition $u\in C([0,T];H^\gamma)$ for $\gamma\in(0,1]$.