In this work a fractional-step DG-FE method for the time-dependent generalized Boussinesq equations is proposed and analysed. The scheme is composed of two steps. In the first step the original problem is reduced into several scalar elliptic equations. An intermediate velocity and temperature are solved simultaneously. Then in the second step, the incompressibility constraint is enforced and velocity is corrected to be discretely divergence free. Moreover, the introduced elliptic term in the correction step enables the imposition of correct Dirichlet boundary conditions at each temporal step, avoiding the artificial boundary layer introduced by classical pressure correction method. DG-FE discretization strategy is utilized, in which the discontinuous Galerkin spacial discretization for flow equations is employed to obtain local mass conservation and traditional finite element spacial discretization is adopted for heat equation to reduce degrees of freedom. By choosing different symmetry and penalty parameters, SIPG-FE and NIPG-FE methods can be utilized. The consistency and stability of both methods are proved. Preliminary error estimates proving the optimal spacial order and suboptimal temporal order are carried out. Based on a different error equation and the preliminary error estimates, the optimal temporal convergence order is obtained. Numerical tests including a benchmark simulating square cavity flow are then presented, to verify the theoretical analysis and validate the method.

在这项工作中，提出并分析了时间相关广义 Boussinesq 方程的分数步 DG-FE 方法。该方案由两个步骤组成。在第一步中，原始问题被简化为几个标量椭圆方程。同时求解中间速度和温度。然后在第二步中，强制执行不可压缩性约束，并将速度校正为无离散散度。此外，校正步骤中引入的椭圆项能够在每个时间步骤中施加正确的狄利克雷边界条件，避免了经典压力校正方法引入的人工边界层。采用 DG-FE 离散化策略，其中流动方程的间断Galerkin空间离散化得到局部质量守恒，热方程采用传统的有限元空间离散化来降低自由度。通过选择不同的对称性和惩罚参数，可以使用 SIPG-FE 和 NIPG-FE 方法。证明了两种方法的一致性和稳定性。执行初步误差估计，证明最佳空间顺序和次优时间顺序。基于不同的误差方程和初步误差估计，获得最佳时间收敛顺序。然后提出了包括模拟方腔流动的基准在内的数值测试，以验证理论分析并验证该方法。