This paper studies discretization of time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Most of the analysis in the literature has been performed on fully-discrete methods using first order methods in time, typically the implicit Euler time integrator. Our aim is to show which kind of error bounds can be obtained using any time integrator, both in the full order model (FOM), applied to compute the snapshots, and in the POD-ROM method. To this end, we analyze in this paper the continuous-in-time case for both the FOM and POD-ROM methods, although the POD basis is obtained from snapshots taken at a discrete (i.e., not continuous) set times. Two cases for the set of snapshots are considered: The case in which the snapshots are based on first order divided differences in time and the case in which they are based on temporal derivatives. Optimal pointwise-in-time error bounds {between the FOM and the POD-ROM solutions} are proved for the $L^2(\Omega)$ norm of the error for a semilinear reaction-diffusion model problem. The dependency of the errors on the distance in time between two consecutive snapshots and on the tail of the POD eigenvalues is tracked. Our detailed analysis allows to show that, in some situations, a small number of snapshots in a given time interval might be sufficient to accurately approximate the solution in the full interval. Numerical studies support the error analysis.

翻译：本文研究使用本征正交分解降阶模型（POD-ROM）对时间依赖偏微分方程（PDE）进行离散化。现有文献中的大多数分析均针对全离散方法展开，且通常采用一阶时间格式（典型的隐式欧拉时间积分器）。本文旨在阐明，在使用任意时间积分器时，无论是用于计算快照的全阶模型（FOM），还是POD-ROM方法，所能获得的误差界形式。为此，本文同时分析了FOM和POD-ROM方法在连续时间情形下的特性，尽管POD基函数是从离散（即非连续）时刻采样获得的快照中构造的。研究考虑了两种快照集情况：基于时间一阶差商的快照，以及基于时间导数的快照。针对半线性反应扩散模型问题，证明了FOM与POD-ROM解之间在$L^2(\Omega)$范数下的最优逐点时间误差界。我们追踪了连续快照间时间间隔以及POD特征值尾部对误差的影响。通过详细分析表明，在某些情形下，给定时间区间内仅需少量快照即可在全区间内获得对解的精确逼近。数值实验验证了误差分析的正确性。