## 4 Conclusions and Discussion  
The depth-averaged continuum method is an effective approach for solving the avalanche dynamics model. It simplifies all the physical complexity and uncertainty of the motion of a large quantity of grains in a \"granular fluid\". Considering the nonlinear properties of the governing equations of the dynamics, developing an accurate and efficient numerical scheme is challenging. We applied the TVD-MacCormack scheme, which was initially used to solve the Saint-Venant equation for shallow water flow. The scheme introduces a TVD term into the MacCormack scheme after the corrector step, enabling second-order spatial and temporal accuracy during the calculation. In addition, the with/without grain algorithm was integrated into the predictor-corrector steps during the computation. The application of the scheme was checked in two scenarios. The first was a granular flume flow striking square obstacles with varying side lengths splitting into two granular streams. Thus, a fan-shaped configuration formed and the opening between the two granular streams grew with the increasing side length. The simulation results generally agreed well with the laboratory measurements with regard to the granular depth upstream  
Figure 11: Comparison of the computational dam break configuration at times \(t=40\) s (**upper**) and \(t=80\) s (**lower**) with open (**left**) and closed (**right**) boundary conditions (unit: m). from the obstacle. Using the TVD-MacCormack scheme, we captured shock waves and observed successive waves appearing in the tail of the avalanche between the two streams of flow during the simulations, as observed in previous laboratory research. In the dam breach simulations, the second scenario, a box-shaped pile of granular mass collapsed following the sudden emergence of a rectangular breach. We changed the width of the breach and tracked the flow rate \(Q\), the velocity \(u\) in the \(x\)-direction, and the granular depth \(h\) during the breaching of the dam. The simulation results indicated that both the flow rate and the velocity in the \(x\)-direction reached large values within a short period of time and then decreased rapidly. The average values and the fluctuations of the relatively steady flow rate and the velocity were also found to decrease with the breach width. The results of the granular dam break simulations also revealed that the boundary conditions (open or closed) affected the collapse of the granular pile, i.e., grains evenly breached out under closed boundary conditions, whereas the granules breaching out of the opening were mostly grains adjacent to the boundaries under open boundary conditions. Although the applied governing equations and the corresponding numerical algorithms have been well developed, the TVD-MacCormack scheme (a second-order spatial and temporal finite-element scheme) was successfully implemented here to solve a two-dimensional dynamics model of a depth-averaged avalanche with high accuracy. In addition, the simulation of a granular flume flow-obstacle interaction scenario and a granular dam breaking scenario captured some valuable features in these two scenarios, i.e., shock waves with a fan-shaped configuration and successive waves in the tail of the avalanche between the two streams in the first scenario, as well as discontinuous granular collapses in the second scenario. Capturing these two complex and typical granular-flow phenomena indicates the applicability and effectiveness of combining the TVD-MacCormack scheme and S-H model to simulate dam breaking and inclined flow-obstacle interaction cases. This paper investigates what happens when a dense granular flow strikes on a rigid obstacle, described by a wall boundary, rather than a topographic feature with a finite slope. It is shown that the numerical method of the TVD-MacCormack scheme has a shock-capturing ability and can be used to solve for the motion of granular dam breaking.