# A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations

## TL;DR

This paper enhances the POD-ROM method for advection-diffusion-reaction equations by introducing a stabilizing post-processing strategy that improves numerical stability and accuracy in strongly advection-dominated regimes.

## Contribution

It proposes a novel a posteriori stabilization technique for POD-ROM, applicable during both snapshot generation and simulation, specifically targeting strongly advection-dominated problems.

## Key findings

- Improved stability in advection-dominated regimes
- Enhanced accuracy of reduced order models
- Effective stabilization in numerical simulations

## Abstract

In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framework and applied to advection-diffusion-reaction problems. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.