With the two-dimensional shallow water equations, we refer to the original model of A. de Saint-Venant (1871). The model is discretized using finite volumes, with an explicit Godunov scheme (MUSCL-EVR reconstruction is also an available option). The original nonlinear equations can be supplemented with source terms such as rain, wind, soil infiltration and friction. Reference: A. de Saint-Venant (1871). Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit. Comptes-rendus hebdomadaires des séances de l’Académie des Sciences, 73 , pp. 148 – 154 (in French).
Two-dimensional shallow water models with porosity are an interesting path for the large-scale modelling of floodplains with urbanized areas. Here, the porosity parameter (defined cellwise) accounts for the reduction in storage due to the presence of buildings and other structures in the floodplain. In comparison with the traditional SWES equations, the SP model provides a faster computational tool for large scale simulations. Reference: V. Guinot & S. Soares-Frazao (2017). Flux and source term discretization in two-dimensional shallow water models with porosity on unstructured grids. Int. J. Numer. Meth. Fluids, 50: 309-345.
The DIP model is established from an integral mass and momentum balance whereby both porosity and flow variables are defined separately for control volumes and boundaries, and a closure scheme is introduced to link control volume-and boundary-based flow variables. With a more accurate definition of porosity than in the SP model, the DIP model also provides a faster computational tool for large scale simulations. It also features an unsteady momentum dissipation model that is essential to reproducing the specific features of some sharp flow transients. Reference: V. Guinot, B. F. Sanders & J. E. Schubert (2017). Dual integral porosity shallow water model for urban flood modelling. Advances in Water Resources, 103: 16-31.
The Local Inertial Approximation to the Shallow Water Equations (LISWEs) is based on a simplification of the momentum flux, whereby the inertial term is neglected. This formulation is often used to simulate flows with small velocities/low Froude numbers. Reference: R. Xia (1994). Impact of coefficients in momentum equation on selection of inertial models. Journal of Hydraulic Research, 32(4), 615–621.
All of the aforementioned models are discretized in conservation form using finite volumes. The hyperbolic part of the governing equations is discretized using explicit, Godunov-type schemes. The first-order Godunov [1] reconstruction is proposed as a default, while the MUSCL-EVR approach [2, 3] is proposed for some of the models. The MUSCL-EVR reconstruction is faster than the more widespread RK2 MUSCL-Hancock technique and exhibits fewer issues in the presence of wetting/drying fronts.
References: [1] S. K. Godunov (1959). A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.), 1959, Volume 89, Number 3, 271–306. [2] B. van Leer (1977). Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. Journal of Computational Physics, Volume 23, Issue 3, March 1977, Pages 276-299. [3] S. Soares-Frazão, V. Guinot (2007). An eigenvector-based linear reconstruction scheme for the shallow-water equations on two-dimensional unstructured meshes. Int. J. Numer. Meth. Fluids, 53 (1), 23-55.