Model Order Reduction (MOR) of systems of non-linear(parameterized) Hyperbolic Partial Differential Equations (PDEs) is still an uncharted territory in the scientific community. Moving discontinuities are representative features of this class of problems and pose a major hindrance to obtain effective reduced-order model representations, since typically bases with high spatial frequency are needed to accurately capture these moving discontinuities. We will discuss a MOR framework to efficiently capture the travelling dynamics of such systems. The motivation of this work is to enable the usage of multi-phase hydraulic models, such as the Drift Flux Model (DFM)  in developing drilling automation strategies for real-time down-hole pressure management.
The DFM is a system of multiscale non-linear PDEs, whose convective subset is conditionally hyperbolic. Convection dominated problems, such as the DFM, admit solutions, which possess a diagonal structure in space-time diagram and high solution variability. As a first step, we apply standard MOR approaches  to obtain a reduced-order representation of the DFM for a representative multi-phase shock tube test case. We capture the dynamics in an essentially non-oscillatory manner but we obtain a small dimensionality reduction. Since the dimension of the reduced model is still too large, we develop new techniques for deriving more efficient alternative reduced-order models for this class of problems.
We invoke the idea of the method of freezing  and combine it with non-linear reduced basis approximations  to develop an efficient reduced-order model representation, which we demonstrate for several benchmark problems. These benchmark problems embody the challenges faced in the reduced-order representation of the DFM. However, the existing MOR framework  lacks consideration of boundary conditions and multiple fronts. The main novelty of this work is in investigating the performance of combined approach of Method of Freezing and reduced basis approximations in dealing with merging (discontinuous) wave-fronts. Finally, we present numerical experiments and discuss the efficacy of above mentioned approach in terms of computational speed up and computational accuracy compared with standard numerical techniques.