Variational approaches to water wave simulations

Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves

“…The vertices on the wave maker are thus evaluated at time t n+1/2 . By using approximations (23a)-(23c) for the first three free surface and potential energy terms in ( 13) and approximations (27) for the kinetic energy and (29) for the wave maker terms in (13), the space discrete principle (13) becomes the following space-time discrete variational principle 0 = δL φn,+ ,…”

“…The stability of our time integration scheme satisfies the standard criterion [15] for the Störmer-Verlet scheme. The construction and stability of such (novel) time discontinuous Galerkin finite element schemes are considered in Gagarina [13]. In addition, we rename φ n,+ l as φ n l .…”

Power optimisation of a rogue-wave energy device in a contraction

“…For nonlinear wave modelling, these models are computationally much faster than the Navier-Stokes equations and well-explored in the wavemodelling applied-mathematics community. For numerical wave modelling, e.g., we refer to boundary/finite-element models of Ma et al [19], Engsig-Karup, Bingham [11] and Boussinesq models [18], the latter which use optimised and reduced vertical resolution (for other variational models see [14], [15], [10]). Typically, second or higher-order time discretisations are used such as compatible modified midpoint ( [8], [10]) or Runge-Kutta schemes.…”

A study of extreme water waves using a hierarchy of models based on potential-flow theory

“…Subsequently, once {η n+1/2 , n+1/2 , q n+1/2 } are determined, these relations are also used to recover the updates. The above procedure is a shortcut from a more extensive time-discrete VP that fully recovers the MMP scheme [32].…”