Post-Processing Reduced-Order Models for Transport-Dominated Problems by Gegenbauer Reconstruction

In this paper, we develop a physics-based post-processing technique for data-driven reduced-order models (ROMs) of transport-dominated problems. Besides the slow decay of the Kolmogorov n-width, ROMs based on globally supported bases often produce unphysical oscillations when approximating solutions with shocks or sharp gradients, a phenomenon analogous to Gibbs oscillations in spectral approximations. To address this issue, we introduce a post-processing framework based on Gegenbauer polynomial reconstruction.
The key idea is to re-project the ROM solution onto a Gegenbauer polynomial basis over each interval of analyticity. Originally developed for spectral approximations, Gegenbauer reconstruction achieves spectral accuracy while effectively suppressing Gibbs oscillations. We extend this technique to data-driven ROMs and consider three representative approaches: Proper Orthogonal Decomposition (POD)-Galerkin ROM, Operator Inference (OpInf), and nonlinear manifold ROMs based on convolutional autoencoders (CAE). Numerical results show that the proposed post-processing consistently removes spurious oscillations and substantially improves solution quality for all three ROMs.
For one-dimensional problems, the method is straightforward to implement once discontinuities are detected. We further develop a practical extension to two-dimensional problems using line-by-line reconstruction in each coordinate direction. Extensive numerical experiments demonstrate that the proposed method reduces errors by up to one or two orders of magnitude for inviscid transport problems and significantly outperforms total variation regularization in both numerical accuracy and the sharp resolution of discontinuities.