Novel Numerical Analysis Methods, Using the WENO and WENO-Z Algorithms, for Combining Observational Data with Model Predictions for Improving Forecasts

“Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations.” — Wikipedia Strong constraint 4D-Variational data assimilation (4D-Var) seeks to find optimal initial condition for computing solutions to appropriate dynamical system equations. The optimal initial condition minimizes the weighted mismatch between (i) the computed initial condition and given “background” data and (ii) the computed state trajectory and observed data. The optimal initial condition and subsequent prediction are sensitive to the numerical method used to discretize the dynamical system equations. Especially, if the expected forecast has large gradients, the sensitivity of a prediction due to a given selected numerical method is significantly more pronounced. Weighted essential non-oscillatory(WENO) schemes, as a class of finite difference schemes, have been applied extensively in computational fluid dynamics for numerically solving problems with solutions which contain both strong shocks and rich smooth region structures. However, WENO, and its improved version WENO-Z, have never been analyzed as discretization techniques for the data assimilation model equations (i.e. data assimilation constraint equations) as compared to existing state-of-the-art discretization strategies. In this dissertation, we analyze the properties of the WENO and WENO-Z schemes, when used in the “4D-Var” process for the linear advection equation, and compare them quantitatively to the following schemes: Upwind Scheme, Lax-Friedrichs Scheme, Lax-Wendroff and Quick Scheme. In our analysis, we also compare the performance of the WENO and WENO-Z based 4D-Var algorithms when used together with L1 norm total variation regularization technique which reduces Gibb’s phenomena of discontinuous observed data in the data assimilation process.