In this paper, we consider the Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equation, usually known as the Cahn-Hilliard-Navier-Stokes (CHNS) system. The CHNS system has been widely embraced to investigate the dynamics of a binary fluid mixture. By utilizing the modified leap-frog time-marching method, we propose a novel numerical algorithm for solving the CHNS system in an efficient and accurate manner. This newly proposed scheme has several advantages. First of all, the proposed scheme is linear in time and space, such that only a linear algebra system needs to be solved at each time-marching step, making it extremely efficient. Also, the existence and uniqueness of numerical solutions are guaranteed for any time step size. In addition, the scheme is unconditionally energy stable with second-order accuracy in time and spectral accuracy in space, such that relatively large temporal and spatial step sizes can be used to obtain reliable numerical solutions. The rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given. Furthermore, we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency. The differences of coarsening dynamics between the Cahn-Hilliard equation and the Cahn-Hilliard-Navier-Stokes equations have been investigated as well.

在本文中，我们考虑了Cahn-Hilliard方程与不可压缩的Navier-Stokes方程，通常称为Cahn-Hilliard-Navier-Stokes（CHNS）系统。CHNS系统已被广泛采用来研究二元流体混合物的动力学。通过使用改进的跨越式跳时方法，我们提出了一种新颖的数值算法，可以高效，准确地求解CHNS系统。该新提出的方案具有多个优点。首先，所提出的方案在时间和空间上是线性的，因此在每个时间步长步骤仅需要求解线性代数系统，从而使其效率极高。同样，对于任何时间步长，数值解的存在性和唯一性都得到了保证。此外，该方案是无条件的能量稳定的，具有时间的二阶精度和空间的频谱精度，因此可以使用相对较大的时间和空间步长来获得可靠的数值解。给出了无条件能量稳定性质以及解存在和唯一性的严格证明。此外，我们提供了几个数值示例来测试所提出的数值算法，并说明其准确性和效率。还研究了Cahn-Hilliard方程和Cahn-Hilliard-Navier-Stokes方程之间的粗化动力学差异。给出了无条件能量稳定性质以及解存在和唯一性的严格证明。此外，我们提供了几个数值示例来测试所提出的数值算法，并说明其准确性和效率。还研究了Cahn-Hilliard方程和Cahn-Hilliard-Navier-Stokes方程之间的粗化动力学差异。给出了无条件能量稳定性质以及解存在和唯一性的严格证明。此外，我们提供了几个数值示例来测试所提出的数值算法，并说明其准确性和效率。还研究了Cahn-Hilliard方程和Cahn-Hilliard-Navier-Stokes方程之间的粗化动力学差异。