Tsunami-HySEA solves the two-dimensional shallow-water system using a high-order (second and third order) path-conservative finite volume method. Values of h, qx and qy at each grid cell represent cell averages of the water depth and momentum components. The numerical scheme is conservative for both mass and momentum in flat bathymetries and, in general, is mass preserving for arbitrary bathymetries. High order is achieved by a non-linear TVD reconstruction operator of the unknowns h, qx, qy and h=h-H. Then, the reconstruction of H is recovered using the reconstruction of h and h. Moreover, in the reconstruction procedure, the positivity of the water depth is ensured. Tsunami-HySEA implements several reconstruction operators: MUSCL (see van Leer, 1979) that achieves second order, the hyperbolic Marquina's reconstruction (see Marquina, 1994), that achieves third order, and a TVD combination of piecewise parabolic and linear 2D reconstructions that also achieves third order (see [21]). The high order time discretization is performed using the second or third order TVD Runge-Kutta method described in Gottlieb and Shu (1998). At each cell interface, Tsunami-HySEA uses Godunov's method based on the approximation of 1D projected Riemann problems along the normal direction to each edge. In particular Tsunami-HySEA implements a PVM-type method that uses the fastest and the slowest wave speeds, similar to HLL method (see [22]). A general overview of the derivation of the high order methods is performed in [23]. For large computational domains and in the framework of TEWS, Tsunami-HySEA also implements a two-step scheme similar to leap-frog for the deep water propagation step and a second-order TVD-WAF flux-limiter scheme, described in [7], for close to coast propagation/inundation step. The combination of both schemes guaranties the mass conservation in the complete domain and prevents the generation of spurious high frequency oscillations near discontinuities generated by leap-frog type schemes. At the same time, this numerical scheme reduces computational times compared with other numerical schemes, while the amplitude of the first tsunami wave is preserved.
Concerning the wet-dry fronts discretization, Tsunami-HySEA implements the numerical treatment described in [1] and [3], that consists of locally replacing the 1D Riemann solver used during the propagation step, by another 1D Riemann solver that takes into account the presence of a dry cell. Moreover, the reconstruction step is also modified in order to preserve the positivity of the water depth. The resulting schemes are well-balanced for the water at rest, that is, they exactly preserve the water at rest solutions, and are second or third order accurate, depending on the reconstruction operator and the time stepping method. Finally, the numerical implementation of Tsunami-HySEA has been performed on GPU clusters [7, 24, 25] and nested-grids implementation available [9, 15, 16, 17, 19, 20]. These facts allow to speedup the computations, being able to perform complex simulations, in very large domains, much faster than real time [9, 15, 16, 17].
The dispersive model implements a formally second-order well-balanced hybrid finite volume/difference (FV/FD) numerical scheme. The non-hydrostatic system can be split into two parts: one corresponding to the non-linear shallow water component in conservative form and the other corresponding to the non-hydrostatic terms. The hyperbolic part of the system is discretized using a PVM path-conservative finite-volume method ([22] and Parés, 2006), and the dispersive terms are discretized with compact finite differences. The resulting ODE system in time is discretized using a TVD Runge-Kutta method (Gottlieb and Shu, 1998).
