In this paper, being based on the idea of dispersion-relation-preserving optimization, a class of high-order high-resolution upwind compact schemes with a free parameter and their consistent boundary scheme are proposed for the solution of the governing equations of the 2D double-diffusive convection. The first derivative and the second derivative, in the vorticity equation, the temperature equation and the concentration equation, are discretized by using the optimal upwind compact scheme on a uniform mesh and the fourth-order symmetrical Padé compact scheme, respectively. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point two dimensional stencil. The fourth-order Runge-Kutta method is utilized for the temporal discretization. To assess numerical capability of the proposed algorithm, particularly its spatial behavior, the problems about the convection diffusion equation, the nonlinear Burgers equation and the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are computed by using the newly proposed algorithm. The numerical results are in excellent agreement with the benchmark solutions and some of the accurate results available in the literature, and show that effectiveness, accuracy and the advantage of better resolution of the present method. After that, steady and unsteady solutions for the double-diffusive convection in a rectangular cavity are also used to assess the efficiency of the present algorithm. The period of oscillation and flow field profiles are in great agreement with the data in the literature for buoyancy ratio, $\lambda =1$. The typical separation and secondary vortices at the bottom corner of the cavity as well as the top corner can be captured well for various buoyancy ratio. These indicate that the present method is suitable for simulating effectively the unsteady double-diffusive convection.

本文基于色散关系保持优化的思想，提出了一类带有自由参数的高阶高分辨率迎风紧凑格式及其一致边界格式，用于求解二维控制方程。双扩散对流。利用均匀网格上的最佳迎风紧致格式和四阶对称Padé紧致格式，分别将涡度方程，温度方程和浓度方程中的一阶导数和二阶导数离散化。使用目前在九点二维模板上构建的四阶紧致差分方案，可以近似得出压力泊松方程。利用四阶Runge-Kutta方法进行时间离散化。为了评估所提出算法的数值能力，特别是其空间特性，利用新方法计算了具有绝热水平壁和垂直加热壁的方腔中对流扩散方程，非线性Burgers方程和自然对流的问题。提出的算法。数值结果与基准解决方案和文献中提供的一些准确结果非常吻合，表明该方法的有效性，准确性和更好分辨率的优势。之后，还使用矩形腔中双扩散对流的稳态和非稳态解来评估本算法的效率。$\lambda =\mathrm{1个}$。对于各种浮力比，可以很好地捕捉到腔底角和顶角处的典型分离涡流和次级涡流。这些表明本方法适合于有效地模拟不稳定的双扩散对流。