Reduced Order Model Closures: A Brief Tutorial

In this paper, we present a brief tutorial on reduced order model (ROM) closures. First, we carefully motivate the need for ROM closure modeling in under-resolved simulations. Then, we construct step by step the ROM closure model by extending the classical Galerkin framework to the spaces of resolved and unresolved scales. Finally, we develop the data-driven variational multiscale ROM closure and then we test it in fluid flow simulations. Our tutorial on ROM closures is structured as a sequence of questions and answers, and is aimed at first year graduate students and advanced undergraduate students. Our goal is not to explain the "how," but the "why." That is, we carefully explain the principles used to develop ROM closures, without focusing on particular approaches. Furthermore, we try to keep the technical details to a minimum and describe the general ideas in broad terms while citing appropriate references for details.