A space-time semi-Lagrangian advection scheme on staggered Voronoi meshes applied to free surface flows

In this article we present a novel space-time semi-Lagrangian advection scheme for the solution of the nonlinear convective terms in hyperbolic conservation laws. The governing equations are discretized on a three-dimensional mesh, composed of a staggered unstructured Voronoi grid on the horizontal plane which is extruded along the vertical direction with z−layers of non-uniform thickness. A high order space-time reconstruction is carried out for the velocity field, that is used for both tracking backward in time the Lagrangian trajectories of the flow and for the interpolation of the transported quantity at the foot of the characteristics. High order in space is achieved via a constrained least-squares reconstruction technique, whereas the ADER procedure is employed for gaining high order of accuracy in time as well. The high order reconstruction polynomials are expanded onto a set of basis functions that are defined in the physical coordinate system for space and in the reference framework for time, thus improving the computational efficiency of the scheme. The trajectory equation of the flow particles is then solved relying on symplectic-type integrators, which are proven to be structure-preserving ODE solvers, unlike standard explicit Runge–Kutta schemes. Application to hydrostatic free surface flows is proposed, demonstrating accuracy and robustness of the novel numerical method via comparison against analytical solutions.