Hydrostatic Model
Uhaina is composed of and hydrodynamic core that solves the non-linear Shallow Water equations (NSW), witten in their pre-balanced formulation [1].
The NSW expresses the dynamics of free surface shallow flows of incompressible fluid. It is an accurate model to simulate long wave hydrodynamics when the vertical acceleration can be neglected and the flow supposed nearly horizontal. In practice it is use for a wide variety of applications such as simulations of large scale storm/hurricane surges, large scale tsunamis propagation, coastal flooding by overflowing.
The pre-balanced formulation reflects the capability of the method to maintain a steady state at rest over time. In fact the gradient of the bathymetry in the source term is not always well balanced with the flux when solved numerically. To prevent this problem and following the work in [1], the ShallowWater system is re-written.
Non-Hydrostatic Model
More recently and still in working progress, Uhaina is composed of an non-hydrostatic module involving the use of the fully nonlinear and weakly dispersive Green-Naghdi equations, in the formulation proposed by [2]. The code implies the two steps solution strategy by [3], allowing an efficient and flexible computation of the dispersive terms: a first step where the non-hydrostatic source term is recovered by inverting the elliptic coercive operator associated to the dispersive effects; a second step which involves the solution of the hyperbolic shallow water system with the source term, computed in the previous phase, which accounts for the non-hydrostatic effects.
References
[1] Q. Liang and A.G.L. Borthwick. “Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography”. Computers and Fluids 38.2 (2009), pp. 221–234 (cit. on p. 4).
[2] D. Lannes and F. Marche. “A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2 D simulations”. Journal of Computational Physics 282 (2014), pp. 238–268 (cit. on pp. 3, 4).
[3] A.G. FIlippini et al. "A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up". Journal of Computational Physics 310 (2016), pp. 381–417.