The principles of optimal control of distributed parameter systems are used to derive a powerful class of numerical methods for solutions of inverse problems, called data assimilation (DA) methods. Using these DA methods one can efficiently estimate the state of a system and its evolution. This information is very crucial for achieving more accurate long term forecasts of complex systems, for instance, the atmosphere. DA methods achieve their goal of optimal estimation via combination of all available information in the form of measurements of the state of the system and a dynamical model which describes the evolution of the system. In this dissertation work, we study the impact of new nonlinear numerical models on DA. High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. The impact of high resolution advection schemes in the framework of inverse problem solution/ DA has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on DA is the subject of this work. In 1-D we use a nonlinear viscous Burgers equation and in 2-D a global nonlinear shallow water model has been used. The results obtained show that using the various advection schemes consistently improves variational data assimilation (VDA) in the strong constraint form, which does not include model error. However, the cost functional included efficient and physically meaningful construction of the background cost functional term, J_b, using balance and diffusion equation based correlation operators. This was then followed by an in-depth study of various approaches to model the systematic component of model error in the framework of a weak constraint VDA. Three simple forms, decreasing, invariant, and exponentially increasing in time forms of evolution of model error were tested. The inclusion of model error provides a substantial reduction in forecasting errors, in particular the exponentially increasing form in conjunction with the piecewise parabolic high resolution advection scheme was found to provide the best results. Results obtained in this work can be used to formulate sophisticated forms of model errors, and could lead to implementation of new VDA methods using numerical weather prediction models which involve high resolution advection schemes such as the van Leer slope limiters and the PPM. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of
the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Spring Semester, 2006. / Date of Defense: April 3, 2006. / Finite Volume Methods, Data Assimilation, Numerical Weather Prediction, Optimal Control, High Resolution Schemes / Includes bibliographical references. / Ionel Michael Navon, Professor Directing Dissertation; James J. O'Brien, Outside Committee Member; Gordon Erlebacher, Committee Member; Qi Wang, Committee Member; Mark Sussman, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_168224 |
| Contributors | Akella, Santharam (authoraut), Navon, Ionel Michael (professor directing dissertation), O'Brien, James J. (outside committee member), Erlebacher, Gordon (committee member), Wang, Qi (committee member), Sussman, Mark (committee member), Department of Mathematics (degree granting department), Florida State University (degree granting institution) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text |
| Format | 1 online resource, computer, application/pdf |
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