In this thesis a numerical study of the one dimensional viscous Burgers equation is conducted. The discretization techniques Finite Differences, Finite Element Method and Group Finite Elements are applied and their impact on model reduction techniques, namely Proper Orthogonal Decomposition (POD), Group POD and the Discrete Empirical Interpolation Method (DEIM), is studied. This study is facilitated by examination of several common ODE solvers. Embedded in this process, some results on the structure of the POD basis and an alternative algorithm to compute the POD subspace are presented. Various numerical studies are conducted to compare the different methods and the to study the interaction of the spatial discretization on the ROM through the basis functions. Moreover, the results are used to investigate the impact of Reduced Order Models (ROM) on Optimal Control Problems. To this end, the ROM is embedded in a Trust Region Framework and the convergence results of Arian et al. (2000) is extended to POD-DEIM. Based on the convergence theorem and the results of the numerical studies, the emphasis is on implementation strategies for numerical speedup. / Master of Science

On the Blue Nile in Sudan, near the Ethiopian border, there is a measurement station named El Deim. The discharge assessments carried out at this station are crucial for the water resource management in Sudan. Due to changed conditions, caused by a heightening of the downstream-located Roseires dam, new methods for discharge assessment are needed. The objective of the present study was to examine possibilities and methodologies to assess the discharge at this station. The flow dynamics was examined through steady state as well as dynamic hydraulic modeling by use of the Mike 11 modeling software package. By simulating possible future scenarios, in the aspect of discharge variations in the Blue Nile and water level variations in the reservoir, the effects from the raised dam on El Deim could be studied. The model was based on bathymetrical data in form of cross sections. As boundary conditions for the simulation, measured and synthetic data series of discharge and water levels were used. The known measured water levels at El Deim were compared with the simulated water levels at El Deim for the same discharge scenarios. The modeled value corresponds well to the measured values. The existing discrepancies between the simulated and measured values are likely caused by insufficient bathymetrical data.   Simulation results show that the flow dynamics at El Deim are highly dependent on the water level of the reservoir and the discharge’s rate of variation. Accordingly, rating curves were created for a range of water levels at the reservoir. With the use of these curves, and tables/equations based on them, the discharge can be rated by knowing the water level at the Roseires dam and El Deim. However, the results from this study are more a description of the principles of how the discharge ratings could be performed. If the methodology and rating tools from this study are planned to be implemented the model must be improved with more bathymetrical data. The improvements are needed to create more accurate curves, tables and equations for discharge rating. Discharge ratings can then be produced and enable better operation of Roseires dam and a more efficient use of the valuable water resources in Sudan. In order to test the applicability of the created model and produced rating tools they should be compared with new measurement data from El Deim with the heightened Roseires dam fully implemented.   It is possible to assess the discharge at El Deim even when backwater effects affect the station. The methodology developed in this thesis would be applicable for similar studies at other locations.