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A central scheme for shallow water flows along channels with irregular geometry

Jorge Balbás, Smadar Karni (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical...

A central scheme for shallow water flows along channels with irregular geometry

Jorge Balbás, Smadar Karni (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical...

A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport

Manuel Jesús Castro Díaz, Enrique Domingo Fernández-Nieto, Tomás Morales de Luna, Gladys Narbona-Reina, Carlos Parés (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity...

A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport

Manuel Jesús Castro Díaz, Enrique Domingo Fernández-Nieto, Tomás Morales de Luna, Gladys Narbona-Reina, Carlos Parés (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different...

A new numerical model for propagation of tsunami waves

Karel Švadlenka (2007)

Kybernetika

A new model for propagation of long waves including the coastal area is introduced. This model considers only the motion of the surface of the sea under the condition of preservation of mass and the sea floor is inserted into the model as an obstacle to the motion. Thus we obtain a constrained hyperbolic free-boundary problem which is then solved numerically by a minimizing method called the discrete Morse semi-flow. The results of the computation in 1D show the adequacy of the proposed model.

A steady-state capturing method for hyperbolic systems with geometrical source terms

Shi Jin (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar...

A steady-state capturing method for hyperbolic systems with geometrical source terms

Shi Jin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar...

About global existence and asymptotic behavior for two dimensional gravity water waves

Thomas Alazard (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds.The proof is based on a bootstrap argument involving L 2 and L ∞ estimates. The L 2 bounds are proved in the paper [5]. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation...

Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System

W. Abou Salem (2012)

Mathematical Modelling of Natural Phenomena

The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large...

An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

François Bouchut, Tomás Morales de Luna (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in...

An entropy-correction free solver for non-homogeneous shallow water equations

Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

An entropy-correction free solver for non-homogeneous shallow water equations

Tomás Chacón Rebollo, Antonio Domínguez Delgado, Enrique D. Fernández Nieto (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

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