## equations

### BibTeX

@MISC{Bernetti_equations,

author = {R. Bernetti and V. A. Titarev and E. F. Toro},

title = {equations},

year = {}

}

### OpenURL

### Abstract

Exact solution of the Riemann problem for the shallow water

### Citations

287 |
Advanced Mathematical Methods for Scientists and Engineers
- Bender, Orszag
- 1999
(Show Context)
Citation Context ...oints. Proof. The stationary points of FL are the roots of the following polynomial P 3 1 (ε) = 2ε 3 − 3ε 2 + (1 − ∆h) 2 (16) To nd approximate values of these roots we use the perturbation technique =-=[15]-=- which gives two meaningful positive roots in the form of a series expansion. The roots are as follows: εsta,1 = 1 − εsta,2 = 1 + √ 2 3 ∆h1/2 − 2 9 √ 2 3 ∆h1/2 − 2 9 ∆h + 7 54 √ 6 ∆h3/2 − 5 243 ∆h2 + ... |

165 | Waves in Fluids - Lighthill - 1978 |

54 | Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32
- Gallouët, Hérard, et al.
- 2003
(Show Context)
Citation Context ...ely. Another approach to the problem is to add an additional equation to the system describing the bottom behavior in time and then try to construct a Riemann solver for the extended system, see e.g. =-=[6]-=-. The idea is that the scheme using such a Riemann solver will not be prone to the problems encountered in conventional advection methods. In this work, we present a new exact solution of the Riemann ... |

34 |
Shock-Capturing Methods for Free-Surface Shallow Flows
- Toro
- 2001
(Show Context)
Citation Context ...n popular due to their ability to treat easily shock waves and contact discontinuities arising in the solution. For a review of modern nite-volume methods as applied to the shallow water equations see=-=[3]-=-. However, in spite of signi cant overall progress made in the eld, serious problems still remain in dealing with geometric source terms arising in the shallow water equations in the case of non-unifo... |

28 |
Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry
- Vazquez-Cendon
- 1999
(Show Context)
Citation Context ...metric source terms. A popular approach to the construction of Godunov-type methods for hyperbolic systems with geometric source terms is to use the so-called upwind discretization of the source term =-=[4, 5]-=-. In the resulting schemes the numerical approximation of the source term is done in such a way as to try to ensure that in the steady-state the ux gradient and the source term are balanced, at least ... |

21 |
Analysis and approximation of conservation laws with source terms
- Greenberg, LeRoux, et al.
- 1997
(Show Context)
Citation Context ... possible conjugate curves ω1. That is, for each right state on the step there are two possible left states UL. Similar non-uniqueness has been observed for other hyperbolic problems by other authors =-=[12, 16]-=-. We now use lemmas 2.1 and 2.5 to reduce non-uniqueness to a more narrow range. Lemma 3.5. In order to satisfy the energy dissipation condition the curve ω2 has to be split into positive and negative... |

15 |
Riemann problems and the WAF method for solving twodimensaional shallow water equations
- Toro
- 1992
(Show Context)
Citation Context ... of a rst-order centred scheme. Good agreement is observed. Next, we illustrate a practical use of the developed exact Riemann solver by incorporating it into the Weighted Average Flux (WAF) 2method =-=[8, 9, 10, 11]-=-, which is a second-order Godunov-type TVD schemes. Though the detailed evaluation of the resulting scheme will be reported in a separate publication, we show some preliminary results which look encou... |

10 |
Exact solutions to the Riemann problem of the shallow water equations with a bottom step
- Alcrudo, Benkhaldoun
- 2001
(Show Context)
Citation Context ...entional advection methods. In this work, we present a new exact solution of the Riemann problem for the shallow water equations with a a discontinuous bottom geometry, which is di erent from that of =-=[7]-=-. The di erence is twofold. Firstly, in our work conservation of mass and momentum are used to derive the Rankine-Hugoniot conditions across the bottom step. Secondly, in order to exclude the multipli... |

3 |
A nite di erence method for the computation of discontinuous solutions of the equations of uid dynamics
- Godunov
- 1959
(Show Context)
Citation Context ...of the geometries encountered in real-life applications much e ort has been made in recent years to develop numerical methods to solve the equations approximately. In particular, Godunov-type methods =-=[2]-=- have proven popular due to their ability to treat easily shock waves and contact discontinuities arising in the solution. For a review of modern nite-volume methods as applied to the shallow water eq... |

3 |
Balancing source terms and ux gradients in high-resolution Godunov methods : the quasi-steady wave-propagation algorithm
- LeVeque
- 1998
(Show Context)
Citation Context ...metric source terms. A popular approach to the construction of Godunov-type methods for hyperbolic systems with geometric source terms is to use the so-called upwind discretization of the source term =-=[4, 5]-=-. In the resulting schemes the numerical approximation of the source term is done in such a way as to try to ensure that in the steady-state the ux gradient and the source term are balanced, at least ... |

1 |
A weighted average ux method for hyperbolic conservation
- Toro
- 1989
(Show Context)
Citation Context ... of a rst-order centred scheme. Good agreement is observed. Next, we illustrate a practical use of the developed exact Riemann solver by incorporating it into the Weighted Average Flux (WAF) 2method =-=[8, 9, 10, 11]-=-, which is a second-order Godunov-type TVD schemes. Though the detailed evaluation of the resulting scheme will be reported in a separate publication, we show some preliminary results which look encou... |

1 |
An e cient solver for near shore ows based on the WAF method. Costal Engineering
- Brocchini, Bernetti, et al.
- 2001
(Show Context)
Citation Context ...e numerical artifacts. We next demonstrate preliminary results of the practical application of present Riemann solver in the framework of Godunov-type upwind methods. Here we use it in the WAF method =-=[8, 9, 3, 10, 11]-=-, which is a second-order TVD schemes. A detailed explanation of the implementation of the Riemann solver in the framework of this method will be reported elsewhere. Fig. 20 shows the preliminary resu... |

1 |
Structure-generated macrovortices and their evolution in very shallow depths
- Brocchini, Mancinelli, et al.
- 2002
(Show Context)
Citation Context ... of a rst-order centred scheme. Good agreement is observed. Next, we illustrate a practical use of the developed exact Riemann solver by incorporating it into the Weighted Average Flux (WAF) 2method =-=[8, 9, 10, 11]-=-, which is a second-order Godunov-type TVD schemes. Though the detailed evaluation of the resulting scheme will be reported in a separate publication, we show some preliminary results which look encou... |

1 |
LeFloch and Mai Duc Thanh. The Riemann poblem for uid ows in a nozzle with discontinuous cross-section
- G
- 2003
(Show Context)
Citation Context ...ts which look encouraging. We note, that in existing literature, there are some Riemann solvers for nonlinear systems with discontinuous geometry which use the formulation with an additional equation =-=[7, 12, 13]-=-. In [7] the authors present a Riemann solver for the shallow water equations with a discontinuous piece-wise constant bottom. The solution is composed of a stationary shock sitting at the bottom disc... |