An adaptive time-stepping scheme is developed for the Zakharov-Rubenchik system to resolve the multiple time scales accurately and to improve the computational efficiency during long-time simulations. The Crank-Nicolson formula and the Fourier pseudo-spectral method are respectively utilized for the temporal and spatial approximations. The proposed numerical method is proved to preserve the mass and energy conservative laws in the discrete levels exactly so that the magnetic field, the density of mass, and the fluid speed are stable on a general class of nonuniform time meshes. With the aid of the priori estimates derived from the discrete invariance and the newly proved discrete Gronwall inequality on variable time grids, sharp convergence analysis of the fully discrete scheme is established rigorously. Error estimate shows that the suggested adaptive time-stepping method can attain the second-order accuracy in time and the spectral accuracy in space. Extensive numerical experiments coupled with an adaptive time-stepping algorithm are presented to show the effectiveness of our numerical method in capturing the multiple time scale evolution for various velocity cases during the interactions of solitons.

为Zakharov-Rubenchik系统开发了一种自适应时间步进方案，以准确解析多个时间尺度并提高长时间仿真过程中的计算效率。分别利用Crank-Nicolson公式和傅里叶伪谱方法进行时间和空间近似。所提出的数值方法被证明能够准确地保留离散级中的质量和能量守恒律，使得磁场、质量密度和流体速度在一般类非均匀时间网格上稳定。借助离散不变性的先验估计和新证明的变时间网格离散Gronwall不等式，严格建立了全离散格式的锐收敛分析。误差估计表明，所提出的自适应时间步进方法可以达到时间上的二阶精度和空间上的频谱精度。广泛的数值实验与自适应时间步进算法相结合，展示了我们的数值方法在捕获孤子相互作用期间各种速度情况的多时间尺度演化方面的有效性。