Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition?

TL;DR

This paper investigates whether including snapshot difference quotients improves the convergence of proper orthogonal decomposition basis functions, finding that their use leads to optimal convergence rates in certain norms.

Contribution

It provides a theoretical and numerical comparison showing that using snapshot difference quotients yields optimal convergence rates in POD basis functions.

Findings

Using difference quotients results in optimal convergence in $C^0(L^2)$ and $C^0(H^1)$ norms.

Without difference quotients, convergence in these norms is suboptimal.

Both approaches achieve optimal convergence in the $L^2(H^1)$-norm.

Abstract

This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be used to generate the proper orthogonal decomposition basis functions? The answer to this question is important, since some published numerical studies use the time difference quotients, whereas other numerical studies do not. The criterion used in this paper to answer this question is the rate of convergence of the error of the reduced order model with respect to the number of proper orthogonal decomposition basis functions. Two cases are considered: the no_DQ case, in which the snapshot difference quotients are not used, and the DQ case, in which the snapshot difference quotients are used. 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 convergence rates in the $L^2(H^1)$-norm are optimal for both the DQ case and the no_DQ case. 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. 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 proper orthogonal decomposition basis functions, one should use the snapshot difference quotients.