High resolution schemes for hyperbolic conservation laws

There is growing concern that flooding is becoming more frequent and severe in Europe. A better understanding of flood regime changes and their drivers is therefore needed. The paper reviews the current knowledge on flood regime changes in European rivers that has traditionally been obtained through two alternative research approaches. The first approach is the data-based detection of changes in observed flood events. Current methods are reviewed together with their challenges and opportunities. For example, observation biases, the merging of different data sources and accounting for nonlinear drivers and responses. The second approach consists of modelled scenarios of future floods. Challenges and opportunities associated with flood change scenarios are discussed such as fully accounting for uncertainties in the modelling cascade and feedbacks. To make progress in flood change research, we suggest that a synthesis of these two approaches is needed. This can be achieved by focusing on long duration records and flood-rich and flood-poor periods rather than on short duration flood trends only, by formally attributing causes of observed flood changes, by validating scenarios against observed flood regime dynamics, and by developing low-dimensional models of flood changes and feedbacks. The paper finishes with a call for a joint European flood change research network.

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Understanding Flood Regime Changes in Europe: A state of the art assessment

Numerical resolution of well-balanced shallow water equations with complex source terms

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

Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography

A numerical technique for the modelling of shallow water flow in one and two dimensions is presented in this work along with the results obtained in different applications involving unsteady flows in complex geometries. A cell-centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured cells is presented. The discretization of the bed slope source terms is done following an upwind approach. In some applications a problem arises when the flow propagates over adverse dry bed slopes, so a special procedure has been introduced to model the advancing front. The applications presented are mainly related with unsteady flow problems

A numerical model for the flooding and drying of irregular domains

A wetting-drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting-drying condition based on steady-state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting-drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests Free surface flow in channels can be described mathematically by the shallow-water system of equations These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two-dimensional dam break flows A cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first- and second-order accuracy Special treatment of the friction term has been adopted and will be described The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, ie that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC-IST) Comparisons of experimental and numerical results are shown

Two‐dimensional dam break flow simulation

Debris flow is modelled using the equations governing the dynamics of a liquid-solid mixture. An upwind finite volume scheme is applied to solve the resulting differential equations in one dimension. These equations have a structure similar to those of the monophasic water flow, differing from them by the presence of some terms characteristic of the bifasic nature of the mixture, such as granular bed erosion velocity, sediment concentration, bed shear stress, etc. The model and the system of equations to be solved are presented with the description of the implementation of the upwind scheme for the resulting hyperbolic conservation system. The numerical method is first order in both space and time. The treatment of the source terms is specified in detail and some comparison with laboratory experiments are presented.

1D Mathematical modelling of debris flow

The two-dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two-dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second-order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case.

The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant–Friedrichs–Lewy (CFL) condition. Copyright © 2007 John Wiley & Sons, Ltd.

P. Brufau

Papers

A numerical model for the flooding and drying of irregular domains

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

Two‐dimensional dam break flow simulation

1D Mathematical modelling of debris flow

The influence of source terms on stability, accuracy and conservation in two‐dimensional shallow flow simulation using triangular finite volumes