Stabilized reduced order models for the advection–diffusion–reaction equation using operator splitting

Reduced order modeling (ROM) coupled with finite element methods has been used effectively in many disciplines to efficiently solve complex problems. However, for advection-dominated flows numerical simulations often contain spurious, nonphysical oscillations which will also be apparent in the ROM simulations. In this work we consider stabilization methods for ROM for the advection–diffusion–reaction (ADR) equation when it is solved both with and without operator splitting. Specifically we consider the streamline-upwind Petrov–Galerkin (SUPG) stabilization method and the spurious oscillations at layers diminishing (SOLD) stabilization method. We build on these methods by constructing a coherent framework which successfully integrates these model reduction, stabilization, and operator splitting approaches, and we provide numerical examples detailing the application of this framework in the ADR setting. The stabilized ROM results are compared numerically with their corresponding full finite element simulations.