In this paper, we prove uniform error bounds for proper orthogonal decomposition (POD) reduced order modeling (ROM) of Burgers equation, considering difference quotients (DQs), introduced in [26]. In particular, we study the behavior of the DQ ROM error bounds by considering $L^2(\Omega)$ and $H^1_0(\Omega)$ POD spaces and $l^{\infty}(L^2)$ and natural-norm errors. We present some meaningful numerical tests checking the behavior of error bounds. Based on our numerical results, DQ ROM errors are several orders of magnitude smaller than noDQ ones (in which the POD is constructed in a standard way, i.e., without the DQ approach) in terms of the energy kept by the ROM basis. Further, noDQ ROM errors have an optimal behavior, while DQ ROM errors, where the DQ is added to the POD process, demonstrate an optimality/super-optimality behavior. It is conjectured that this possibly occurs because the DQ inner products allow the time dependency in the ROM spaces to make an impact.

翻译：本文证明考虑差商（引入自文献[26]）的Burgers方程本征正交分解（POD）降阶建模（ROM）的一致误差界。具体而言，我们通过考虑$L^2(\Omega)$和$H^1_0(\Omega)$的POD空间以及$l^{\infty}(L^2)$和自然范数误差，研究DQ ROM误差界的行为。我们给出若干有意义的数值测试以检验误差界的行为。基于数值结果，在ROM基保持的能量方面，DQ ROM误差比无DQ ROM误差（即采用标准方式构造POD而不使用差商方法）低数个数量级。此外，无DQ ROM误差呈现最优行为，而DQ ROM误差（在POD过程中加入差商）则表现出最优/超最优行为。推测该现象可能源于差商内积使时间依赖性在ROM空间中产生影响。