Current Project: Data assimilation for compressible flows
Data assimilation (DA) aims at bridging the gap between experimental approaches and numerical simulations for studying the behavior of an evolutionary system. In fact, DA techniques have long been used as the main tool for combining observational data and outputs of numerical models to provide a more realistic estimate of the evolving state of a dynamical system. These techniques have widely been applied in many disciplines for example in calibrating atmospheric models as well as for weather and ocean forecasting. In this project, we are going to provide a solid mathematical ground for some of DA techniques including 3D-Var and 4D-Var when applied to compressible flow models and in general hyperbolic conservation laws (HCLs). However, solutions of HCLs are known to produce shock discontinuities in finite time which poses numerous difficulties from both theoretical and computational points of view. To cope with these difficulties in the analysis of numerical methods for HCLs, in particular, discontinuous Galerkin method, one idea is the optimal reconstruction of the numerical solution. In this project, we aim at bringing this idea into the numerical analysis of DA techniques for compressible flows in order to alleviate the difficulties stemming from the hyperbolic nature of governing equations.