## 4 Conclusions and Discussion
The granular flow-obstacle interaction case uses a rectangular cylinder as an obstacle, which is not the classical circular shape. This is of practical interest for the design of defending structures that can withstand such avalanches. In addition, it can represent natural obstacles such as trees. The dam breaking scenario may provide basic physical insight into the problem in which a vertical retaining wall breaks, and thus the granular cliff collapses gradually. Simulated discontinuous granular mass collapses are consistent with the observation on the failure process of a natural slope. The absence of an obstacle generates adjacent shock waves, at which point there are rapid changes in the granular flow thickness and flow velocity. Shock waves arise naturally from the hyperbolic structure of the Saint-Venant equations. In diluted granular flows, shock waves high in density develop and the diluted flow can be seen as compressible flow. This paper investigates dense granular flow with a low solid volume fraction that travels past obstacles, and the flow is modelled as an incompressible flow. Conceptualization, [PERSON]; methodology, J.F. ; software, J.F. ; validation, A.P. ; investigation, M.W. ; resources, S.L. ; data curation, Z.C. ; writing--original draft preparation, J.F. ; writing--review and editing, J.F. ; visualization, G.H., L.C. ; supervision, H.Z. ; project administration, H.Z. ; funding acquisition, H.Z. All authors have read and agreed to the published version of the manuscript. This study was funded by the National Natural Science Foundation of China (NSFC) under grant nos. 52178339, 52090084, and 51938008, and the Open Funding of the State Key Laboratory for Track Technology of High-Speed Railways (2021 YJ114). Not applicable. Data Availability Statement: The data that support the findings of this study are available from the corresponding author, [PERSON], upon reasonable request. The authors declare no conflict of interest. * _[PERSON] and [PERSON] (2014)_[PERSON]; [PERSON] [PERSON] A new total variation diminishing implicit nonstandard finite difference scheme for conservation laws. _Comput. Methods Differ. Equ._**2014**, \(2\), 91-98. * _[PERSON] et al. (2018)_[PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Application of finite difference schemes to 1D St. venant for simulating weir overflow. _MATEC Web Conf._**2018**, _147_, 03011. [CrossRef]
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