Adaptive Numerical Methods for Large Scale Simulations and Data Assimilation

Numerical simulation is necessary to understand natural phenomena, make assessments and predictions in various research and engineering fields, develop new technologies, etc. New algorithms are needed to take advantage of the increasing computational resources and utilize the emerging hardware and software infrastructure with maximum efficiency. Adaptive numerical discretization methods can accommodate problems with various physical, scale, and dynamic features by adjusting the resolution, order, and the type of method used to solve them. In applications that simulate real systems, the numerical accuracy of the solution is typically just one of the challenges. Measurements can be included in the simulation to constrain the numerical solution through a process called data assimilation in order to anchor the simulation in reality. In this thesis we investigate adaptive discretization methods and data assimilation approaches for large-scale numerical simulations. We develop and investigate novel multirate and implicit-explicit methods that are appropriate for multiscale and multiphysics numerical discretizations. We construct and explore data assimilation approaches for, but not restricted to, atmospheric chemistry applications. A generic approach for describing the structure of the uncertainty in initial conditions that can be applied to the most popular data assimilation approaches is also presented. We show that adaptive numerical methods can effectively address the discretization of largescale problems. Data assimilation complements the adaptive numerical methods by correcting the numerical solution with real measurements. Test problems and large-scale numerical experiments validate the theoretical findings. Synergistic approaches that use adaptive numerical methods within a data assimilation framework need to be investigated in the future. This work was supported in part by the National Science Foundation through the awards NSF CAREER ACI–0413872, NSF CCF-0515170, NSF ITR AP&IM 0205198, NSF CCF 0515170, by NOAA, by NASA, and by the Houston Advanced Research Center through the award H59/2005. To my wife and my parents