Projection-based reduced order models (ROM) based on the weak form and the strong form of the discontinuous Galerkin (DG) method are proposed and compared for shock-dominated problems. The incorporation of dissipation components of DG in a consistent manner, including the upwinding flux and the localized artificial viscosity model, is employed to enhance stability of the ROM. To ensure efficiency, the discrete empirical interpolation method (DEIM) is adopted to enable hyper-reduction, for which the upwinding flux is decomposed into the central part and the dissipation part. The maximum local wave speed in the upwinding dissipation part is compressed and approximated using the DEIM approach, and the same strategy is applied to the artificial viscosity. Energy stability is proved with the strong-form-based ROM prior to hyper-reduction for the one-dimensional scalar cases. Eigenvalue spectrum is analyzed to verify and compare the stability properties of the two proposed ROMs. Several benchmark cases are conducted to test the performance of the proposed models. Results show that stable computations with reasonable acceleration for shock-dominated cases can be achieved with the ROM built on the strong form.

提出了基于不连续伽辽金（DG）方法的弱形式和强形式的基于投影的降阶模型（ROM），并对冲击控制问题进行了比较。以一致的方式结合 DG 的耗散分量，包括上风通量和局部人工粘度模型，用于提高 ROM 的稳定性。为保证效率，采用离散经验插值法（DEIM）实现超约化，将逆风通量分解为中心部分和耗散部分。上风消散部分的最大局部波速使用DEIM方法压缩和近似，同样的策略应用于人工粘度。对于一维标量情况，在超归约之前使用基于强形式的 ROM 证明了能量稳定性。分析特征值谱以验证和比较两种提议的 ROM 的稳定性特性。进行了几个基准案例来测试所提出模型的性能。结果表明，对于以强形式构建的 ROM，可以在冲击主导的情况下实现具有合理加速度的稳定计算。