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：在本文中,我们证明,考虑到[ 26] 中引入的差异商数(DQs),对汉堡方程式的正正正或正方形分解(POD)降序建模(ROM)有统一的错误界限。特别是,我们通过考虑$L2(\Omega)美元和$H1_0(Omega)美元(Omega)美元(POD)空间和$L ⁇ infty}(L2/2)美元和自然中下调差错。我们提出了一些有意义的数字测试,以检查误差界限的行为。根据我们的数字结果,DQROM误差是比无DQ差(POD是按标准方式构建的,即不采用DQ方法)的数级小的数级。此外,没有DQQROM错误具有最佳行为方式,而DQ误差(将DQ添加到POD进程时,显示一种最佳/超优性能度行为。它预测,因为DQ产品允许在内部撞击时段发生这种撞击。