Energy-based Modeling and Simulation of Shallow Water Waves in a Tube with Moving Boundary

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

“…Within this section, energy-conserving approximations of the model equations are derived, employing the principle of least action. The main motivation for these discretization schemes is the following result borrowed from [9] and being mainly a restatement of the earlier result [10] in material-fixed coordinates.…”

“…In [8], a family of approximated finite-dimensional models is proposed to preserve the port-Hamiltonian structure of open systems models, utilizing conservative generalized leapfrog schemes with consistent orders and a finite volume perspective on staggered grids. This actual contribution deals with higher-order approximation schemes for the system considered in [9], i.e., a boundary-actuated 1D shallow-water model with moving boundary and arbitrary cross-section. To cope with the timevarying spatial domain, the model equations are formulated using material-fixed coordinates, which are also known as Lagrange coordinates.…”

“…As the distributed shallow-water model can be derived using the principle of least action (cf. [9,10]), the action functional and the Lagrange functional constitute the starting point also for the derivation of the lumped approximation, i.e., the lumped parameter model. In particular, the Lagrange functional is discretized w.r.t.…”

“…Due to the fact, that the conservation of mass is taken into account as an auxiliary condition, the resulting model is a system of semi-explicit differential-algebraic equations (DAE). In contrast to [9], within the actual contribution, higher-order quadrature formulae are employed. Although this is expected to deliver more accurate numerical approximations, the scheme results in more involved nonlinear DAE.…”

“…This actual contribution deals with higher‐order approximation schemes for the system considered in [9], i.e., a boundary‐actuated 1D shallow‐water model with moving boundary and arbitrary cross‐section. To cope with the time‐varying spatial domain, the model equations are formulated using material‐fixed coordinates, which are also known as Lagrange coordinates.…”