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Assessment of sensitivity of depth integrated solutions to longitudinal discontinuities on the channel bed

Mohammadi, Ramez
Depth integrated equations can be easily solved over large domains to provide flood inundation maps. In urban and rural areas however, there may be numerous natural or artificial bottom boundary discontinuities in the form of rapid variations in bed elevation. Such discontinuities cause abrupt changes in the source terms of the governing equations and can significantly affect stability and accuracy of the numerical solution. This study concentrates on the assessment of sensitivity of the governing equations to longitudinal discontinuities and proposes solutions to alleviate associated numerical complications. The presence of dry regions in the domain is also considered which requires additional modifications in the code to deal with the wet/dry fronts. In this thesis, Godunov’s type Finite Volume Method (FVM) is used for the numerical solution of the shallow water equations. Weighted Averaged Flux (WAF) method, that is 2nd order extension of the first order Godunov’s scheme, based on HLL Riemann solver, is used to compute the fluxes. The source term treatment is based on the “hydrostatic reconstruction” and first and second order well-balanced schemes are obtained. Creation of probable negative water depths during linear piecewise reconstruction of the water surface near dry areas has been prevented in the 2nd order well-balanced scheme. Eight test cases with their available analytical solutions that are widely used in the literature are selected to validate the developed codes. The results of the developed codes are also compared with sets of experimental data if available. Test cases are solved by pure WAF method without well-balancing property, WAF method with first order well-balanced scheme, and WAF method with second order well-balanced scheme. Two test cases are specifically selected to show the capability of the developed codes in solving flows in regions with wet and dry areas. In the end, a new test case is introduced to observe behavior of the numerical solutions in a bed consisting of a series of sharp corners due to positive and negative steps. It is observed that well-balanced schemes can produce water surface profiles without spurious oscillations and steady-state horizontal hydrostatic water surface is recovered without noisy fluctuations.