Quantitative comparison of variational and sequential data assimilation techniques for one-dimensional initial-value problems of ideal MHD

State-of-the-art predictions of the solar-wind and space weather phenomena are today largely based on the equations of magnetohydrodynamics (MHD). Despite their sophistication and success, the forecasting potential of global MHD models is often undermined by uncertainties in model inputs; the initial and boundary conditions are generally not known and must be estimated. This study therefore investigates the use of data assimilation strategies to minimize forecast errors in the context of initial-value problems of the one-dimensional ideal MHD equations. Several canonical MHD wave propagation problems involving both smooth and discontinuous solutions, including those having strongly non-linear behaviour with shocks, are considered in a set of twin experiments with varying synthetic observational data sparsity. Two data assimilation strategies are quantitatively compared, namely the Ensemble Kalman Filter (EnKF) and strong-constraint variational data assimilation. For the latter, the necessary adjoint model is derived, summarized, and validated. The study represents the first use of variational data assimilation applied to ideal magnetohydrodynamics and demonstrates its potential advantages over sequential approaches. In particular, for the numerical experiments considered herein, it is found that the variational approach consistently achieved superior performance and stability compared to the EnKF method. In addition, two different strategies for mitigating data assimilation induced errors associated with violation of the divergence-free property of the magnetic field are introduced and assessed. Finally, the present study provides the technical background and quantitative justification for future investigations of variational data assimilation aimed at enhancing three-dimensional simulations of the solar wind and space weather processes.