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

TL;DR

This work develops a fully discrete, filtered Lie-Trotter splitting for the Zakharov system on the torus that operates directly on the original system without auxiliary variables. By extending discrete Bourgain spaces to the ZS and establishing discrete multilinear estimates for Schrödinger–wave interactions, the authors prove convergence under very low regularity, obtaining an error of order $\mathcal{O}(\tau^{s/2}+N^{-s})$ for $s\in(0,2]$. The analysis shows the scheme approximately preserves the system's geometric structure in the fully discrete setting, and numerical experiments confirm the theoretical convergence rates and good conservation properties over long times. These results bridge the gap between continuous well-posedness theory and practical fully discrete schemes in low-regularity regimes, with implications for simulations of Langmuir-wave dynamics in higher 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 $\mathcal{O}(τ^{s/2}+N^{-s})$ 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.

Figures (8)

• Figure 1: The $H^{1/2}\times L^2\times H^{-1}$ error of the unfiltered Lie splitting scheme for the one-dimensional ZS with rough initial data $(E_0, z_0, z_1)\in H^{s_1+1/2}\times H^{s_1}\times H^{s_1-1}$.
• Figure 2: The $H^{1/2}\times L^2\times H^{-1}$ error of the fully discretized filtered Lie splitting scheme for the one-dimensional ZS with rough initial data $(E_0, z_0, z_1)\in H^{s_1+1/2}\times H^{s_1}\times H^{s_1-1}$.
• Figure 3: The $H^{1/2}\times L^2\times H^{-1}$ error of the filtered Lie splitting scheme for the two-dimensional ZS with rough initial data $(E_0, z_0, z_1)\in H^{s_1+1/2}\times H^{s_1}\times H^{s_1-1}$.
• Figure 4: The $H^1\times H^{1/2}\times H^{-1/2}$ error of the filtered Lie splitting scheme for the three-dimensional ZS with rough initial data $(E_0, z_0, z_1)\in H^{s_1+1}\times H^{s_1+1/2}\times H^{s_1-1/2}$.
• Figure 5: The $H^1\times H^{1/2}\times H^{-1/2}$ error of the filtered Lie splitting scheme for the three-dimensional ZS with rough initial data $(E_0, z_0, z_1)\in H^{s_1+1}\times H^{s_1+1/2}\times H^{s_1-1/2}$ with $s_1=0.25$. Left: numerical errors computed against a reference solution with $N=2^8$ and $\tau=2^{-15}$. Right: numerical errors computed against a reference solution with $N=2^9$ and $\tau=2^{-17}$.

Theorems & Definitions (28)

• Definition 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• proof
• Lemma 3.4
• Theorem 3.5
• Lemma 4.1
• proof