On the sensitivity and accuracy of proper-orthogonal-decomposition-based reduced order models for Burgers equation

• Published in: Computers & fluids 2015-01, Vol.106, p.19-32
• Main Authors: Behzad, Fariduddin, Helenbrook, Brian T., Ahmadi, Goodarz
• Format: Article
• Language: English
• Subjects: Burgers equation; Computational fluid dynamics; Computer simulation; Eigenvalue problem; Eigenvalues; Error detection; Errors; Mathematical models; Proper orthogonal decomposition; Reduced order; Reduced order modeling; Round-off error; Stabilization
• ISSN: 0045-7930; 1879-0747
• DOI: 10.1016/j.compfluid.2014.09.041

•Bounds are derived for the effect of numerical errors of the POD (Proper Orthogonal Decomposition) eigenvalues.•Numerical errors of the POD eigenvectors are investigated.•Estimates are given for the number of snapshots required for “snapshot” methods.•The effects of mode quality, stabilization, and dimension on the ROM (Reduced Order Modeling) are examined.•The so called issues are investigated for Burgers equation. Two aspects of proper-orthogonal-decomposition-based reduced order modeling (POD-ROM) of the Burgers equation are examined. The first is the sensitivity of the eigenvalue spectrum and POD modes to round-off errors and errors caused by using a reduced number of snapshots in the POD. For both the direct and the snapshot method of solving the POD problem, solutions obtained using LAPACK’s DGEEV are compared to a new method that we call the “deflation” method. The deflation method always gives positive eigenvalues where as LAPACK often gives spurious negative eigenvalues. However, the direct method using DGEEV is the only method that gives POD modes that are orthogonal to machine precision. Error estimates from linear algebra are used to explain this and also to show that the POD converges with second-order accuracy in the number of snapshots. The minimum number of snapshots needed to obtain a reasonable eigenvalue spectrum is estimated. In the second part of the paper, the effects of mode quality, ROM stabilization, and ROM dimension are investigated for low- and high-Reynolds number simulations of the Burgers equation. The ROM error is assessed using two errors, the error of projection of the problem onto the POD modes (the out-plane error) and the error of the ROM in the space spanned by POD modes (the in-plane error). The numerical results show not only is the in-plane error bounded by the out-plane error (in agreement with theory) but it actually converges faster than the out-of-plane error. The total error is only weakly affected by the quality and orthogonality of the POD modes. Stabilization of the ROM has a positive effect at high-Re, but when the underlying grid used to derive the ROM is well-resolved, stabilization is not necessary.