In a recent work (Koc et al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the authors showed that including difference quotients (DQs) is necessary in order to prove optimal pointwise time error bounds for proper orthogonal decomposition (POD) reduced models of heat equation. this work, we introduce new approach DQs POD procedure. Instead computing modes using all snapshot data and DQs, only ...