## 4 Conclusions  
This study developed an improved model to replicate the complete sequence of events, encompassing the hydrodynamic phenomena in the reservoir area, the progression of dam breach, and the spread of flooding in the downstream inundation area. The model comprises four modules: breach initiation assessment with or without infiltration, vertical erosion of the breach, lateral development of the breach, and slope stability. Furthermore, it is seamlessly incorporated with a two-dimensional flood model that is based on open-source technology and makes use of several central processing units (CPUs) and graphics processing units (GPUs). Our salient contribution can be summarized as: (1) In the evolution model of the breach, the critical starting velocity formulas with and without seepage influence are established. (2) Based on the theoretical analysis of the initial movement of slope sediment, the lateral evolution formula for calculating the width of the breach is derived. (3) The iteration of each time step is realized by directly updating the DEM of the computational grid, and no special processing of the computational grid flux is needed. (4) The upstream reservoir area is included in the whole computational domain to participate in the calculation, and the water depth of the reservoir area in each time step reflects the change in the water volume in the reservoir area, which can simulate the two-dimensional hydrodynamic process of the reservoir area. At the same time, the water depth and flow velocity of the previous time step are used as the driving force for the evolution of the outburst in the next time step, avoiding the calculation process of the outburst flow by using the weir formula. The reliability, adaptability, and efficiency of the integrated coupling model are confirmed through comparison analysis and sensitivity analysis, using the Taum Sauk dam and Tangjiashan Barrier Lake as application cases. The obtained results demonstrate strong concurrence with the measured data. In the Taum Sauk dam, the relative errors of the four important parameters, the peak discharge of the breach, the top width of the final breach, the depth of the breach, and the arrival time of the maximum peak discharge are all within \(\pm\)10%. Although the relative error of the completion time of the final breach is greater than 10%, it is about 30% less than the relative error of the physical model. However, the model presented in this paper has three limitations. Firstly, it does not take into account the erosion process of the downstream slope during the breach evolution. Secondly, the model is only capable of simulating a single breach in the dam and cannot simulate multiple breaches. Thirdly, the model cannot simulate piping failure. In the future, we will carry out physical tests to verify the model parameters. We will enhance the physical process of the breach evolution model and expand its range of applicability. Simultaneously, the further examination of various breaches and the superimposition of breaches in the dam should be conducted to verify that the simulation scenarios and outcomes closely resemble reality. Conceptualization, H.L. ; methodology, H.L. and Z.W. ; software, H.L. ; validation, H.L., Z.W. and D.Z. ; formal analysis, D.Z. ; investigation, H.L. and Z.W. ; resources, D.Z. ; data curation, H.L. and D.Z. ; writing--original draft preparation, H.L. and Z.W. ; writing--review and editing, D.Z. and L.X. ; visualization, Z.W. ; supervision, D.Z. and L.X. ; project administration, D.Z. and L.X. ; funding acquisition, D.Z. All authors have read and agreed to the published version of the manuscript. This study was supported by the National Key Research and Development Program of China (2022 YFC3005501), the MWR Major Science & Technology Program (SKS-2022007) and the IWHR Research & Development Support Program (WH0145B022021, JZ110145B0062024). Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request.