Uniform Bounds with Difference Quotients for Proper Orthogonal Decomposition Reduced Order Models of the Burgers Equation

TL;DR

This paper establishes uniform error bounds for POD reduced order models of the Burgers equation using difference quotients, showing improved accuracy and optimality compared to standard methods through numerical validation.

Contribution

It introduces a novel approach incorporating difference quotients into POD ROMs, resulting in significantly smaller errors and enhanced optimality in modeling the Burgers equation.

Findings

DQ ROM errors are several orders of magnitude smaller than noDQ errors.

DQ approach leads to super-optimality in error bounds.

Numerical tests confirm the theoretical error bounds and improved accuracy.

Abstract

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.