Developments in survey technology such as light
detection ranging and laser-scanning are able to provide
high-resolution topography data sets. In shallow water equations based
modeling, the use of high-resolution topography data is generally
desirable, because it is considered to be a more accurate
representation of the ``real world''. Indeed, high-resolution
discretization of topography leads to more accurate representation of
preferential flow paths and obstructions which influence the global
flow behaviour inside the domain.
However, the integration of these data into the numerical model is often challenging because of finite computer resources. The challenge comes from the scale difference between the computational domain and the topographical features. If each topographical feature was discretized explicitly, the cell number of the resulting mesh and consequently the simulation wall-time would be unfeasibly high. Instead of explicitly discretizing the small-scale topography, its influence can be conceptually accounted for on coarser meshes to reduce the computational cost. These approaches are commonly referred to as ``coarse grid approaches'' or ``scaling approaches''.
Aim of this doctoral thesis is the development of coarse grid approaches for the shallow water model. Hereby, two approaches will be investigated: (1) friction-law based coarse grid approach, (2) porosity-based coarse grid approach. The approaches allow simulations on a coarser resolution while still maintaining an acceptable accuracy. The development of such approaches is of interest in engineering applications, such as the fast prediction of flood inundation areas in case of a fast flood wave or the relatively new field of physical modeling based catchment hydrology.
The friction-law based coarse grid approach uses an artificially increased roughness coefficient, two additional calibration parameters that describe the geometry of the topographical features and the so-called ``inundation ratio'', which is the ratio between water depth and roughness height. Using automated calibration, good agreement between the scaled shallow water model and high-resolution reference solutions and measurement data was achieved.
Further, in this thesis a porosity-based coarse grid approach was developed, which enables full inundation of unresolved features by means of water depth-dependent porosity terms. A Godunov-type method for the solution of the equations was developed, whereby the reconstruction of cell values at the cell interfaces was identified as a source of spurious oscillation. A monotonicity treatment was suggested to address this issue.
The friction-law based as well as the porosity-based coarse grid approach yield results with comparable accuracy to high-resolution classical shallow water models for water depths. However, flow velocities can not be reproduced with the same accuracy. Further, processes at subgrid-scale can not be reproduced.
The benefit of the developed approaches was demonstrated in this work. In the investigated cases, utilizing coarse grid approaches reduced the wall-time of the simulation 2 up to 3 orders of magnitude.