| In this dissertation, we introduce several strategies to improve the efficiency of the Krylov deferred correction (KDC) methods for special structured ordinary and partial differential equations with algebraic constraints. We first study the semi-implicit KDC (SI-KDC) technique which splits stiff differential equation systems into different components and applies different low-order time marching schemes to these components. Compared with the fully implicit KDC (FI-KDC) method, our analysis and preliminary numerical results for differential algebraic equations show that the SI-KDC schemes are more efficient due to the reduced number of operations in each spectral deferred correction (SDC) iteration. Next, we apply the SI-KDC scheme to simulate a two-scale model describing the mass transfer processes in drinking water treatment applications, in which some set of chemical species move from one distinct phase to a second distinct phase. We also present an improved effective model to further advance the efficiency of the multiscale modeling. Finally, we investigate the parareal method to parallelize the KDC techniques, and present some preliminary numerical results to show its potential in large scale simulations. |
