ARE THE SNAPSHOT DIFFERENCE QUOTIENTS NEEDED IN THE PROPER ORTHOGONAL DECOMPOSITION?

This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients (DQs) of the snapshots be used to generate the proper orthogonal decomposition (POD) basis functions? The answer to this question is important, since some published numerical studies use the time DQs, whereas other numerical studies do not. The criterion used in this paper to answer this question is the optimality of the convergence rate of the error of the reduced order model with respect to the number of POD basis functions. Since to the best of our knowledge a definition of the optimality of the convergence rate is not available, we propose one in Definition 3.1 in this paper. Two cases are considered: the no_DQ case, in which the snapshot DQs are not used, and the DQ case, in which the snapshot DQs are used. For each case, two types of POD bases are used: the L-2-POD basis, in which the basis is generated in the L-2-norm, and the H-1-POD basis, in which the basis is generated in the H-1-norm. The error estimates suggest that the convergence rates in the C-0(L-2)-norm and in the C-0(H-1)-norm are optimal for the DQ case, but suboptimal for the no DQ case. The theoretical investigation, which uses two numerically validated assumptions on the POD projection error and the POD Ritz projection error, suggests the following convergence rates: In the DQ case, the error estimates are optimal in all norms (i.e., the C-0(L-2)-norm, the C-0(H-1)-norm, and the L-2(H-1)-norm) for both the L-2-POD basis and the H-1-POD basis. In the no DQ case, however, the error estimates are suboptimal in the C-0(L-2)-norm for the L-2-POD basis and in the C-0(H-1)-norm for both the L-2-POD basis and the H-1-POD basis. Numerical tests are conducted on the heat equation and on the Burgers equation. The numerical results support the conclusions drawn from the theoretical error estimates. For both the heat equation and the Burgers equation, and for all norms and bases considered, the convergence rates for the DQ case are much higher than (and usually twice as high as) the corresponding convergence rates for the no DQ case. Overall, the theoretical and numerical results strongly suggest that, in order to achieve optimal pointwise-in-time rates of convergence with respect to the number of POD basis functions, one should use the snapshot DQs.