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 \( h^{4}+\tau ^{2\alpha ^{\dagger }/3} \) 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.

我们对具有与声速成反比的无量纲参数ε ∈ (0,1]的 Zakharov 系统 (ZS) 的四阶紧致有限差分格式的误差界和守恒​​定律进行了严格分析。极限状态，即0 < ε ≪ 1，由于ZS中波算子的扰动和初始数据的不相容性，解具有高度振荡的波和出射初始层。解以O ( ε ) 波长随时间传播，O (1/ ε ) 空间速度，以及O ( ε ² ) 和O(1) 准备充分和准备不足的初始数据的振幅。高振荡给 ZS 数值方法的误差界限分析带来了明显的困难。在这项工作中，h是网格大小，τ是时间步长，我们给出了一个统一的误差界限\ ( h^{4}+\tau ^{2\alpha ^{\dagger }/3} \)以及准备不足的初始数据和误差界限h ⁴ / ε + τ ² / ε ³对于使用包括能量方法和截止技术在内的工具准备不足的初始数据。紧凑方案提供了比一般二阶方法更好的空间分辨率，并大大降低了计算成本。还提供了数值模拟来证实我们的理论分析。