Mountainous streams are characterized by highly variable bed topography, steep gradients, and small depths. As a result, modeling of the flow in these channels can be challenging. The accuracy and stability of the model depend on the discretization, but there are little existing criteria or guidelines for specifying the discretization. When the depth becomes small compared to the discretization scale, the source terms generally dominate in the Saint-Venant equations. However local variations can lead to sub-critical/supercritical transitions necessitating a conservative upwind shock-capturing numerical scheme. A previous Fourier analysis of such a scheme applied to the linearized non-dimensional 1D Saint-Venant equations showed that the oscillations in the steady state solution depend on the amplitude of the bed profile perturbation and the ratio of the discretization scale to a characteristic length scale. The characteristic length scale is defined as the ratio of the uniform flow depth to the average bed slope. In this paper, the definition of the slope and the depth in the parameters and the utility of the parameters are investigated in the case of two-dimensional (2D) finite element flow model. The last non-dimensional parameter is used as a mesh refinement indicator with an idealized test case, flow past a submerged groin. The parameters are valid for any numerical scheme and are relatively easy to calculate and implement. The analysis will lead us to a stable and accurate hydrodynamic solution which will help with better solutions to other models, e.g. morphology models, transport models, habitat models, ice models etc.
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