Fitting methods in the application of nonlinear hyperbolic equations

Introduced in this study is the fitting method in the computational aerodynamics field, which, through the motion grid, tracks the boundary between regions with mass and massless regions in the flow field, establishing an algorithm that accurately describes the flow field near this boundary. The domain does not include massless regions with singularity, and the computational grid is distributed only in regions with mass. The main work of this paper expands the original shock fitting method to shallow water equations, using weak solution descriptions of flow behavior near the wet/dry front based on local Riemann problems to address geometrically induced errors caused by grid coordinate trans-formations, the DEER algorithm based on discrete equivalent equations is introduced to eliminate errors in spatial metric discretization, achieving volume conservation law in finite difference methods. By comparing the new algorithms with thin-film algorithms using examples of dam break over a flat plane on dry beds and dry-bed generation, the research results show that on coarser grids, the new algorithm can capture accurate velocity fields near the wet/dry front that thin-film algorithms cannot discern, with higher resolution and computational efficiency compared to conventional methods.