Control‐oriented models of the shallow water equations using energy‐conserving discretization schemes

In this study, we introduce higher‐order approximation schemes for a 1D shallow‐water model with a moving boundary and arbitrary cross‐section. The model equations are formulated using Lagrange coordinates to handle the time‐varying spatial domain. By discretizing the action functional on a material‐fixed grid and applying an appropriate quadrature scheme, we derive a finite‐dimensional model. This model, taking mass conservation into account as an auxiliary condition, results in a system of semi‐explicit differential‐algebraic equations (DAE). Unlike previous work, we employ higher‐order quadrature formulae to enhance numerical accuracy, albeit at the cost of more complex nonlinear DAE. In order to compare the performance of the resulting models obtained from using different quadrature schemes, a comprehensive simulation study is conducted.