J. M. González-Vida

NUMERICAL TREATMENT OF WET/DRY FRONTS IN SHALLOW FLOWS WITH A MODIFIED ROE SCHEME

This paper deals with the analysis of some numerical difficulties related to the appearance of wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids. The fluid is supposed to be governed by the Shallow Water equations and the discretization of the equations is performed, when wet/dry fronts do not appear, by means of the Q-scheme of Roe upwinding the source terms introduced in Ref. 40. This scheme is well-balanced in the sense that it solves exactly stationary solutions corresponding to water at rest. Wet/dry fronts cannot be correctly treated with this scheme: it can produce negative values of the thickness of the fluid layer and stationary solutions corresponding to water at rest including wet/dry transitions are not exactly solved. In Refs. 3–5 some variants of this numerical scheme have been proposed that partially solve these difficulties. Here we propose a new variant: at intercells where wet/dry transitions occur, a Nonlinear Riemann Problem is considered instead of a Linear one. The exact solutions of these nonlinear problems, which are easy to calculate, are used in order to define the numerical fluxes. We investigate the properties of the resulting scheme and present some comparisons with the numerical results obtained with some other modified numerical schemes proposed previously.

Numerical Treatment of the Loss of Hyperbolicity of the Two-Layer Shallow-Water System

This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an ‘optimal’ amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.

Improved FVM for two-layer shallow-water models: Application to the Strait of Gibraltar

This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of Roes Q-scheme are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [Vazquez-Cendon ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J Comp Phys 1999;148:497-526; Garcia-Navarro P, Vazquez-Cendon ME. On numerical treatment of the source terms in the shallow water equations. Comput Fluids 2000;29(8):17-45; Castro MJ, Macias J, Pares C. A Q-Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. Math Model Numer An 2001;35(1):107-27]. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data and some tests on friction sensitivity are also presented.

Performance Benchmarking of Tsunami-HySEA Model for NTHMP’s Inundation Mapping Activities

The Tsunami-HySEA model is used to perform some of the numerical benchmark problems proposed and documented in the “Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop”. The final aim is to obtain the approval for Tsunami-HySEA to be used in projects funded by the National Tsunami Hazard Mitigation Program (NTHMP). Therefore, this work contains the numerical results and comparisons for the five benchmark problems (1, 4, 6, 7, and 9) required for such aim. This set of benchmarks considers analytical, laboratory, and field data test cases. In particular, the analytical solution of a solitary wave runup on a simple beach, and its laboratory counterpart, two more laboratory tests: the runup of a solitary wave on a conically shaped island and the runup onto a complex 3D beach (Monai Valley) and, finally, a field data benchmark based on data from the 1993 Hokkaido Nansei-Oki tsunami.

The Lituya Bay landslide-generated mega-tsunami. Numericalsimulation and sensitivity analysis

The 1958 Lituya Bay landslide-generated mega-tsunami is simulated using the Landslide-HySEA model, a recently developed finite volume Savage-Hutter Shallow Water coupled numerical model. Two factors are crucial if the main objective of the numerical simulation is to reproduce the maximal run-up, with an accurate simulation of the inundated area and a precise re-creation of the known trimline of the 1958 mega-tsunami of Lituya Bay. First, the accurate reconstruction of the initial slide. Then, the choice of a suitable coupled landslide-fluid model able to reproduce how the energy released by the 5 landslide is transmitted to the water and then propagated. Given the numerical model, the choice of parameters appears to be a point of major importance, this leads us to perform a sensitivity analysis. Based on public domain topo-bathymetric data, and on information extracted from the work of Miller (1960), an approximation of Gilbert Inlet topo-bathymetry was set up and used for the numerical simulation of the mega-event. Once optimal model parameters were set, comparisons with observational data were performed in order to validate the numerical results. In the present work, we demonstrate that a shallow water type 10 of model is able to accurately reproduce such an extreme event as the Lituya Bay mega-tsunami. The resulting numerical simulation is one of the first successful attempts (if not the first) at numerically reproducing in detail the main features of this event in a realistic 3D basin geometry, where no smoothing or other stabilizing factors in the bathymetric data are applied.

A High-Order Finite Volume Method for Nonconservative Problems and Its Application to Model Submarine Avalanches

In this chapter we investigate how to apply a high-order finite volume method to discretize the model proposed in [FeBo08] to study submarine avalanches.

Simulation of Internal Waves in the Strait of Gibraltar Using a Two-layer Shallow-water Model

This paper is concerned with the simulation of the flow of a stratified fluid through a channel with irregular geometry. The channel is supposed to have a straight axis and to be symmetric with regard to a vertical plane passing through its axis. The cross-sections are supposed to be of arbitrary shape. The fluid is assumed to be composed of two shallow layers of immiscible fluids of constant density. Moreover, we assume that the flow is one dimensional, i.e., at every layer the velocities are uniform over the cross-section and the thickness only depend on the coordinate related to the axis and on the time. Therefore, the equations to be solved are a system of two coupled Shallow Water equations with source terms involving depth and breadth functions. Extensions of the Q-schemes of van Leer and Roe are proposed where the coupling and source terms are treated by adapting the techniques developed in [9], [5] and [2]. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar, performing simulations of tidal effects and comparing the results with observed data.

Chapter 25 – A parallel 2D finite volume scheme for solving the bilayer shallow-water system: Modellization of water exchange at the Strait of Gibraltar

Publisher Summary This chapter presents a parallel 2D finite volume scheme for solving the bilayer shallow-water system, with a focus on modellization of water exchange at the Strait of Gibraltar. In the Strait of Gibraltar, which connects the Atlantic Ocean with the Mediterranean Sea, two layers of different waters can be distinguished: (1) the colder and less saline Atlantic water flowing at surface and penetrating into the Mediterranean, and (2) the deeper, denser Mediterranean water flowing into the Atlantic. In this chapter, a 2D two-layer shallow water system is presented together with an explicit edge-based finite volume numerical scheme. Due to the size of the domain, a parallel implementation based on object oriented MPI (OOMPI) is performed. The domain decomposition is carried out by using Chaco as a mesh partitioner integrated in the code. A special treatment of the open boundaries is implemented to reduce the communication among processors. The numerical model is applied to the simulation of the flow trough the Strait of Gibraltar with real bathimetric and coastline data. The measurements of the performance of the parallel implementation are also discussed.