Structure-preserving reduced order models for conservation laws with source terms

Numerical simulation of partial differential equations is fundamental for studying complex physical phenomena. However, the high computational cost of High-Fidelity (HF) methods, such as Finite Element (FE) and Finite Volume (FV) schemes, makes them prohibitively expensive for parametrized problems in multi-query contexts. Reduced Order Models (ROMs) address this issue through dimensionality reduction, maintaining reasonable accuracy. Nevertheless, this promising approach shows significant limitations in the context of hyperbolic conservation laws, where classical ROMs often fail due to the presence of discontinuities and spurious oscillations that generate physically inadmissible values (e.g., negative density or water height). This work addresses these challenges by introducing an alternative framework named the collocated Reduced Order Model (cROM), which differs from common projection-based model (pROM). We investigate strategies for preserving the structure of conservation laws, with a specific focus on positivity and conservation properties, in the context of ROMs. To guarantee positivity, the cROM is combined with two transformations that are positivity-preserving by construction: the Logarithmic-Exponential (LE) and the Square-Root (SR). Numerical analysis demonstrates that the SR transformation provides satisfactory results for the linear transport equation and shows strong applicability for the shallow water equations, whereas the LE transformation exhibits limitations that require further investigation. For the conservation, the introduction of a specific offset on standard cROM allows us to theoretically achieve such property. The numerical validation of this approach exhibits significant improvements for both linear advection and shallow water equations. In conclusion, this study demonstrates that a suitable transformation, such as the square-root, can effectively preserve the positivity of physical variables at the numerical level, opening the way for applying these techniques to 2D or 3D problems. The recovery of the conservation property on cROM, while still yielding preliminary results, represents a promising area for future development.