Residual-Based Stabilized Reduced-Order Models of the Transient Convection–Diffusion–Reaction Equation Obtained Through Discrete and Continuous Projection

Galerkin and Petrov–Galerkin projection-based reduced-order models (ROMs) of transient partial differential equations are typically obtained by performing a dimension reduction and projection process that is defined at either the spatially continuous or spatially discrete level. In both cases, it is common to add stabilization to the resulting ROM to increase the stability and accuracy of the method; the addition of stabilization is particularly common for convection-dominated systems when the ROM is under-resolved. While continuous and discrete approaches can be equivalent in certain settings, a plethora of different techniques have emerged for each approach. However, to the best of our knowledge, a thorough comparison of these techniques is currently missing. In this work, we take a first, foundational step and provide an in-depth review of seven commonly used residual-based ROM stabilization strategies within the setting of finite element method (FEM) discretizations using the convection-dominated convection–diffusion–reaction (CDR) equation, an established testbed for stabilization methods. We present the formulations in a unified setting, highlight connections between the strategies, and numerically assess the strategies. In the spatially continuous case, we examine the Galerkin, streamline upwind Petrov–Galerkin (SUPG), Galerkin/least-squares (GLS), and adjoint (ADJ) stabilization methods. For the GLS and ADJ methods, we examine formulations constructed from both the “discretize-then-stabilize” technique and the space–time technique. In the spatially discrete case, we examine the Galerkin, least-squares Petrov–Galerkin (LSPG), and adjoint Petrov–Galerkin (APG) methods. We summarize existing analyses for these methods and provide numerical experiments, comparing competing methods for the first time in the literature and assessing the impact of stabilization parameters and time step sizes. Our numerical experiments demonstrate that residual-based stabilized methods developed via continuous and discrete processes yield substantial improvements over standard Galerkin methods when the underlying FEM model is under-resolved. We find that SUPG, space–time GLS, and space–time ADJ are the best continuous stabilization techniques considered. For discrete ROMs, we find that APG is more accurate than LSPG at the expense of a smaller region of stability with respect to the stabilization parameter. The combination of an APG ROM constructed on top of a SUPG FEM is the overall best performing method. The review, discussion, and numerical inter-comparison of the seven stabilizations strategies using the CDR equations serves as a stepping stone toward a comprehensive investigation and further development of stabilization methods for more challenging problems.