Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

We present rigorous analysis on the error bound and conservation laws of a fourth-order compact finite difference scheme for Zakharov system (ZS) with a dimensionless parameter ε ∈ (0,1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0 < ε ≪ 1, the solutions have highly oscillatory waves and outgoing initial layers due to the perturbation from wave operator in ZS and the incompatibility of the initial data. The solutions propagate with O(ε) wavelength in time, O(1/ε) speed in space, and O(ε2) and O(1) amplitudes for well-prepared and ill-prepared initial data respectively. The high oscillation brings noticeable difficulties in analyzing the error bounds of numerical methods to the ZS. In this work, with h the mesh size and τ the time step, we give a uniform error bound h4+τ2α‡/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ h^{4}+\tau ^{2\alpha ^{\dagger }/3} $\end{document} for the well- and less-ill-prepared initial data and an error bound h4/ε + τ2/ε3 for the ill-prepared initial data with tools including energy methods and cut-off techniques. The compact scheme provides much better spatial resolution than general second-order methods and reduces the computational cost a lot. Numerical simulations are also provided to confirm our theoretical analysis.