Regularization Techniques for Reduced Order Models in Convection-Dominated Flows

This dissertation aims to provide an overview of our novel stabilization strategy for Reduced Order Models (ROMs) applied to convection-dominated flows in an under-resolved regime: the Approximate Deconvolution Leray ROM (ADL-ROM). In the last decades, Galerkin Reduced Order Models (G-ROMs) have gained significant attention within the scientific community due to the ability to drastically reduce the computational cost of numerical simulations, an essential goal for real-world applications. Despite their success, the use of ROMs in convection-dominated under-resolved regime has been hindered by numerical instability. Specifically, while the total energy remains bounded, the standard G-ROM leads to inaccurate and unreliable solutions. This issue highlights the necessity of ROM stabilization techniques. One widely adopted approach to address this challenge is the Regularized ROMs (Reg-ROMs), which apply explicit spatial filtering to stabilize G-ROMs. A well-known example is the Leray ROM (L-ROM). However, existing Reg-ROMs often introduce excessive numerical diffusion in their attempt to mitigate G-ROM instabilities, potentially compromising solution accuracy. To overcome these limitations, the new ADL-ROM integrates Approximate Deconvolution (AD) operators into the classical Leray ROM framework, enhancing accuracy without sacrificing numerical stability. Specifically, we propose two ADL-ROM strategies based on Lavrentiev and van Cittert deconvolution methods. Our numerical investigations demonstrate that, for relatively large filter radii, ADL-ROM is significantly more stable than the standard G-ROM, and more accurate than the L-ROM. Additionally, ADL-ROM exhibits reduced sensitivity to model parameters in comparison to L-ROM. Furthermore, we prove a priori error bounds for both the AD operator and the ADL-ROM. To our knowledge, these results represent the first numerical analysis of approximate deconvolution in a ROM context. Then, we discuss ongoing research meant to extend the ADL-ROM methodology to parameter spaces. Numerical and theoretical results are illustrated through several computational simulations of convection-dominated flows.