A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

We present a novel staggered semi-implicit hybrid FV/FE method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type FV method on the staggered grid. Next, a classical continuous FE scheme provides the free surface elevation at the vertex of the primal mesh. The semi-implicit strategy followed circumvents the contribution of the surface wave celerity to the CFL-type time step restriction making the proposed algorithm well-suited for low Froude number flows. The conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme is able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and validated via numerical experiments. The solution of several Riemann problems attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions and experimental data

A numerical model for the simulation of a solitary wave in a coastal region

In this paper we propose a numerical model for the simulation of the tsunami wave propagation on coastal region. The model can simulate the wave transformation due to refraction, shoaling, diffraction and breaking phenomena that take place in the surf zone and can simulate the wet front progress on the mainland. The above mentioned model is based on the numerical integration of the Fully Non-linear Boussinesq Equations in the deep water region and of the Non-linear Shallow Water Equations in the surf zone. These equations are expressed in an integral contravariant formulation and are integrated on generalized curvilinear boundary conforming grid that can reproduce the complex morphology of the coast line. The numerical integration of the model equations is implemented by a high order Upwind WENO numerical scheme that involves an exact Riemann Solver. For the simulation of the wet front progress on the dry bed, the exact solution of the Riemann problem for the wet-dry front is used. The capacity of the proposed model to simulate the wet front progress velocity is tested by numerical reproducing the dam-break problem on a dry bed. The capacity of the proposed model to correctly simulate the tsunami wave evolution and propagation on the coastal region is tested by numerical reproducing a benchmark test case about the tsunami wave propagation on a conic island

Contravariant Boussinesq equations for the simulation of wave transformation, breaking and run-up

We propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and near shore currents in computational domains representing the complex morphology of real coastal regions. The motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non- linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume–finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock- capturing finite volume scheme; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is applied on a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite Pescara harbour (Italy)

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed