Low-regularity error estimates of a filtered Lie-Trotter splitting scheme for the Zakharov system in arbitrary dimensions

Abstract

In this paper, we establish error estimates for a fully discrete, filtered Lie splitting scheme applied directly to the Zakharov system -- a model whose solutions may exhibit extremely low regularity in arbitrary dimensions. Remarkably, we find that the scheme exhibits an \emph{approximately structure-preserving} behavior in the fully discrete setting. Our error analysis relies on multilinear estimates developed within the framework of discrete Bourgain spaces. Specifically, we prove that if the exact solution $(E,z,z_t)$ belongs to $H^{s+r+1/2}\times H^{s+r}\times H^{s+r-1}$, then the numerical error measured in the norm $H^{r+1/2}\times H^{r}\times H^{r-1}$ is of order $\mathcal{O}(\tau^{s/2}+N^{-s})$ for $s\in(0,2]$, where $r=\max(0,\tfrac d2-1)$ and $N$ denotes the number of spatial grid points. To the best of our knowledge, this is the first rigorous error estimate for splitting methods applied directly to the original Zakharov system -- without introducing auxiliary variables for reformulating the equations. Such reformulations typically compromise the system's intrinsic geometric structure, whereas our approach preserves it approximately by operating on the system in its native form. Finally, we present numerical experiments that corroborate and illustrate the theoretical convergence rates.