http://www.stccmop.org/CORIE/modeling/selfe/SELFEpaper.pdf

SELFE: A semi-implicit Eulerian–Lagrangian finite-element model for cross-scale ocean circulation

This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large-scale inundations. Copyright © 2003 John Wiley & Sons, Ltd.

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

https://web.stanford.edu/~fringer/publications/obf_oceanmodelling_2006_14.pdf

An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator

Shock-capturing non-hydrostatic model for fully dispersive surface wave processes

https://scholarworks.wm.edu/cgi/viewcontent.cgi?article=1802&context=vimsarticles

Seamless cross-scale modeling with SCHISM

Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems

High resolution methods for multidimensional advection diffusion problems in free-surface hydrodynamics

A finite volume discretization of the mixed form of Richards' equation leads to a nonlinear numerical model which yields exact local and global mass conservation. The resulting nonlinear system requires sophisticated numerical strategies, especially in a variable saturated flow regime. In this paper a nested, Newton-type algorithm for the discretized Richards' equation is proposed and analyzed. With a judicious choice of the initial guess, the quadratic convergence rate is obtained for any time step size and for all flow regimes.

A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form

Iterative solutions of mildly nonlinear systems

New test cases for frictionless, three-dimensional hydrostatic flows have been derived from some known analytical solutions of the two-dimensional shallow water equations. The flow domain is a paraboloid of revolution and the flow is determined by the initial conditions, the nonlinear advective terms, the Coriolis acceleration and by the hydrostatic pressure. Wetting and drying is also included. Some specific properties of the exact solutions are discussed under different hypothesis and relative importance of the forcing terms. These solutions are proposed for testing the stability, the accuracy and the efficiency of numerical models to be used for simulating environmental hydrostatic flows. The computed solutions obtained with a semi-implicit finite difference-finite volume algorithm on unstructured grid are compared with the corresponding analytical solutions in both two and three space dimension. Excellent agreement are obtained for the velocity and for the resulting water surface elevation. Comparison of the computed inundation area also shows a good agreement with the analytical solution with degrading accuracy observed when the inundation area becomes relatively large and for long simulation time.

Paola Zanolli

Papers

Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems

High resolution methods for multidimensional advection diffusion problems in free-surface hydrodynamics

A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form

Iterative solutions of mildly nonlinear systems

Comparing analytical and numerical solution of nonlinear two and three‐dimensional hydrostatic flows