Data Assimilation of Compressible Flows With Discontinuities: Evaluating Algorithms on Sod's Shock Tube

Data assimilation (DA) combines noisy observations with uncertain model predictions to obtain optimal state estimation. It has been used extensively in numerical weather prediction and is increasingly used in computational fluid dynamics. However, the application of DA to compressible flows with discontinuities such as shocks or detonation fronts is far less explored. In this paper, we examine three different DA algorithms applied to 1D, non‐reacting, compressible flows: The particle filter (PF), the ensemble Kalman filter (EnKF), and 4D‐Var. The Sod's shock tube problem is employed as a canonical test case. While the sequential DA methods (PF and EnKF) are able to successfully assimilate sparse pressure measurements, this comes at the risk of smearing sharp gradients due to reconstructing the state as an ensemble average. On the other hand, the 4D‐Var method, applied in the context of a small parameter inverse problem, preserves sharp gradients within the resolution of the forward solver, but may require many iterations to converge to the truth. This study therefore provides assessments of sequential and variational DA methods in 1D shock tube problems and contributes towards applying DA to more complex shock‐laden flows (e.g., in higher dimensions, or reacting flows).