CNAM Paris - France
Reduced Order Models (ROMs) have been introduced to overcome the prohibitive computational cost of numerical simulations in multi-query contexts and complex physical phenomena, while maintaining an acceptable accuracy. In the presence of discontinuities, the standard projection-based ROM (pROM) often fails to preserve intrinsic physical properties of conservation laws, such as conservation and positivity. In this talk, we present a nonlinear transformation that is positivity-preserving by construction. This strategy is included within two different frameworks: a hyper-reduced pROM [1] and a novel collocation ROM (cROM) [2]. These methods rely on the optimal selection of discretisation cells, referred to as collocated points, while employing distinct reduced operators. To illustrate the proposed approach, we perform numerical simulations considering one-dimensional equations, such as the linear advection equation and the shallow water equations. Future work will investigate conservation property and the preservation of steady-state solutions.
[1] M. Bergmann, M. G. Carlino, A. Iollo Model Order Reduction Using a Collocation Scheme on Chimera Meshes: Addressing the Kolmogorov-Width Barrier. SIAM Journal on Scientific Computing, vol 47, 2025.
[2] M. G. Carlino, A. Del Grosso, A. Iollo, D. Sipp Analysis of Collocation-based Model Order Reduction, hal open science, 2026.