Energy-Conserving Discretization of the One-Dimensional Shallow Water Equations in Material-Fixed Coordinates

A generalized approach to the spatial discretization of the one-dimensional shallow water equations with moving boundary and an arbitrary cross-section is developed. Material-fixed coordinates are used to effectively cope with the moving boundary. The methodology involves the discretization of the Lagrangian on a material-fixed grid and the application of different quadrature schemes to derive finite-dimensional models. The proposed scheme explicitly considers mass conservation as an additional constraint, resulting in systems of semi-explicit differential-algebraic equations (DAEs). The particular structure of these DAEs depends on the chosen quadrature scheme and therefore requires slightly different methods for the numerical implementation. These methods are discussed on the basis of three examples, which are compared in simulation studies.