This work aims at employing the discontinuous Galerkin (DG) methods for solving the nonstationary conduction–convection model, which is motivated by advantages of DG methods. The discontinuous element pair $P_k$-$P_{k-1}$-$P_k$ in the space discretization and backward Euler scheme in time are applied, as well as an explicit upwind scheme based on Picard iteration is introduced for the coupled term. Existence and uniqueness of the approximation solutions are proved, along with preserving the unconditional stability of the numerical schemes. The priori error estimates for velocity, pressure and temperature are also derived. Numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the DG methods in simulating practical problems.

这项工作旨在采用非连续Galerkin（DG）方法来解决非平稳传导对流模型，这是由DG方法的优势所推动的。不连续元素对$P_{ķ}$--$P_{ķ-\mathrm{1个}}$--$P_{ķ}$在空间离散化和时间上向后的欧拉方案被应用，并且为耦合项引入了基于Picard迭代的显式迎风方案。证明了逼近解的存在性和唯一性，同时保持了数值格式的无条件稳定性。还可以得出速度，压力和温度的先验误差估计。提供数值测试以验证理论分析并表明DG方法在模拟实际问题中的效率。