UTBEST3D

The numerical solution algorithm in UTBEST3D considers the system of hydrostatic primitive equations (1) with a free surface. A prismatic mesh (Fig. 1) is obtained by projecting a given triangular mesh in the vertical direction to provide a continuous piecewise linear representations of the topography and of the free surface. The vertical columns are then subdivided into layers. If a bottommost prism is degenerate, it is merged with the one above it. Due to the discontinuous nature of the approximation spaces, no constraints need to be enforced on the element connectivity. Hanging nodes and mismatching elements are allowed and have no adverse effects on stability or conservation properties of the scheme. This flexibility with regard to mesh geometry is exploited in several key parts of the algorithm: vertical mesh construction in areas with varying topography (Fig. 4), local mesh adaptivity (Fig. 2), wetting/drying (Fig. 3).

\partial_t {\bf u}_H +\nabla \cdot \left( {\bf u}_H {\bf u} – D_{\bf u} \nabla{\bf u} \right) + \nabla_H P -f_C {\bf k} \times {\bf u}_H =0
\partial_t T + \nabla \cdot \left({\bf u} T – D_T \nabla T \right) = 0
\partial_t S + \nabla \cdot \left({\bf u} S – D_S \nabla S \right) = 0
\partial_t \xi + \nabla_H \cdot \int_{z_b}^{\xi} {\bf u} dz =0\hspace{45mm}(1)
\nabla \cdot {\bf u} =0
\partial_z P= – \frac{\rho}{\rho_0}g
\rho = \rho(T,S,z)

\nabla_H = (\partial_x,\partial_y), {\bf k} = (0,0,1)^T, D_{X, X \in \{{\bf u},S,T\}} – eddy viscosity/diffusivity, \xi – free surface elevation, z_b – bathymetry, {\bf u}= (u,v,w) – velocity, {\bf u}_H= (u,v), T – temperature, S – salinity, P – pressure, \rho – density, \rho_0 – reference density, g – gravity acceleration, f_C – Coriolis coefficient.

Our DG discretization is based on the local discontinuous Galerkin method (Cockburn, Shu, 1998) that represents a direct generalization of the cell-centered finite volume method, the latter being just the piecewise constant DG discretization. One of the features of this method is a much smaller numerical diffusion exhibited by the linear and higher order DG approximations compared to the finite difference or finite volume discretizations. The method guarantees the element-wise conservation of all primary unknowns including tracers, supports an individual choice of the approximation space for each prognostic and diagnostic variable, demonstrates excellent stability properties, and possesses proven local adaptivity skills (Fig. 2).