Non-Linear Shallow Water Equations numerical integration on curvilinear boundary-conforming grids

An Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the Shallow Water Equations on generalized curvilinear coordinate systems is proposed. The Shallow Water Equations are expressed in a contravariant formulation in which Christoffel symbols are avoided. The equations are solved by using a high-resolution finite-volume method incorporated with an exact Riemann Solver. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities on generalized boundary-conforming grids is presented; this procedure allows the numerical scheme to satisfy the freestream preservation property on highly-distorted grids. The capacity of the proposed model is verified against test cases present in literature. The results obtained are compared with analytical solutions and alternative numerical solutions

Time domain simulations of dynamic river networks

The problem of simulating a river network is considered. A river network is considered to comprise of rivers, dams/lakes as well as weirs. We suggest a numerical approach with specific features that enable the correct representation of these assets. For each river the flow of water is described by the shallow water equations which is a system of hyperbolic partial differential equations and at the junctions of the rivers, suitable coupling conditions, viewed as interior boundary conditions are used to couple the dynamics. A different model for the dams is also presented. Numerical test cases are presented which show that the model is able to reproduce the expected dynamics of the system. Other aspects of the modelling such as rainfall, run-off, overflow/flooding, evaporation, absorption/seepage, bed-slopes, bed friction have not been incorporated in the model due to their specific nature

Discontinuous Galerkin Method for 1D Shallow Water Flow with Water Surface Slope Limiter

A water surface slope limiting scheme is applied to numerically solve the one dimensional shallow water equations with bottom slope source term. The total variation diminishing Runge-Kutta discontinuous Galerkin finite element method with slope limiter schemes based on water surface and water depth are investigated for solving one-dimensional shallow water equations. For each slope limiter, three different Riemann solvers based on HLL, LF, and Roe flux functions are used. The three different solvers with slope limiters based on water surface and water depth are applied to simulate idealized dambreak problem, hydraulic jump, quiescent flow, subcritical flow, supercritical flow, and transcritical flow. The proposed water surface based slope limiter scheme is easy to implement and shows better conservation property compared to the slope limiter based on water depth for the tests. Of the three flux functions, the Roe approximation provides the best results while the LF function proves to be least suitable when used with either slope limiter scheme

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com

Modeling Shallow Water Flows on General Terrains

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by performing depth integration along a local direction normal to the bottom surface, and propose a rigorous derivation of this approximation and its numerical solution as an essential step for the future development of the full "cross-flow" integration procedure. We start by defining a local coordinate system, anchored on the bottom surface to derive a covariant form of the Navier-Stokes equations. Depth integration along the local normals yields a covariant version of the shallow water equations, which is characterized by flux functions and source terms that vary in space because of the surface metric coefficients and related derivatives. The proposed model is discretized with a first order FORCE-type Godunov Finite Volume scheme that allows implementation of spatially variable fluxes. We investigate the validity of our SW model and the effects of the bottom geometry by means of three synthetic test cases that exhibit non negligible slopes and surface curvatures. The results show the importance of taking into consideration bottom geometry even for relatively mild and slowly varying curvatures

Efficient explicit finite volume schemes for the shallow water equations with solute transport

This work is concerned with the design and the implementation of efficient and novel numerical techniques in the context of the shallow water equations with solute transport, capable to improve the numerical results achieved by existing explicit approaches. When dealing with realistic applications in Hydraulic Engineering, a compromise between accuracy and computational time is usually required to simulate large temporal and spatial scales in a reasonable time. With the aim to improve the existent numerical methods in such a way to increase accuracy and reduce computational time. Three main contributions are envisaged in this work: a pressure-based source term discretization for the 1D shallow water equations, the analysis and development of a Large Time Step explicit scheme for the 1D and 2D shallow water equations with source terms and the numerical coupling between the 1D and the 2D shallow water equations in a 1D-2D coupled model. The first improvement roughly consists of exploring the pressure and bed slope source terms that appear in the 1D and 2D shallow water equations to discretize them in an intelligent way to avoid extremely reductions in the time step size. On the other hand, the implementation of a Large Time Step scheme is carried out. In order to relax the stability condition associated to explicit schemes and to allow large time step sizes, reducing consequently the numerical diffusion associated to the original explicit scheme. Finally, two 1D-2D coupled models are developed. They are demonstrated to be fully conservative and are able to approximate well the results obtained by a fully 2D model in terms of accuracy, while the computational effort is clearly reduced. All the advances are analysed by means of different test cases, including not only academic configurations but also realistic applications, in which the numerical results achieved by the new numerical techniques proposed in this work are compared with the conventional approaches

HIGH ORDER SHOCK CAPTURING SCHEMES FOR HYPERBOLIC CONSERVATION LAWS AND THE APPLICATION IN OPEN CHANNEL FLOWS

Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows