## On the Computation of Roll Waves (2000)

Venue: | Math. Model. Num. Anal |

Citations: | 6 - 3 self |

### BibTeX

@ARTICLE{Jin00onthe,

author = {Shi Jin and Yong Jung Kim},

title = {On the Computation of Roll Waves},

journal = {Math. Model. Num. Anal},

year = {2000},

volume = {35},

pages = {463--480}

}

### OpenURL

### Abstract

incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation u(x; 0) = u 0 (x); which arises as a weakly nonlinear approximation of the shallow water equations. The main difficulty associated with the numerical approximation of this problem is its linear instability. Numerical round-off error can easily overtake the numerical solution and yields false roll wave solution at the steady state.

### Citations

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and Nonlinear Waves
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(Show Context)
Citation Context ...x = 0 ; (hv) t + \Gamma hv 2 + 1 2 gh 2 \Delta x = ghS \Gamma C f v 2 ; where g is the gravitational constant, S is the slope of the river bottom and C f is the friction coefficient. It is long known =-=[Whi]-=- that the initial value problem of the above system is linearly stable if (1.2) S ! 4C f ; namely, when the model describes flow in a river bed where the slope S is usually small compared to the frict... |

459 |
Numerical Methods for Conservation Laws
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(Show Context)
Citation Context ...mation of the steady state solution (3.2) at cell [x j \Gamma1=2 ; x j+1=2 ]. This method will be referred to as the interface method. In order to define the numerical flux, we use the Godunov method =-=[LeV]-=-: (3.11) U n j+1=2 = 8 ? ! ? : U n j ; if U n j + U n j+1s0; U n j+1 ; if U n j + U n j+1 ! 0; 0 ; U n j ! 0; U n j+1 ? 0: computation of roll waves 9 We solve (1.5) with initial data being the cell a... |

455 |
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Citation Context ...re considered in a distribution sense. Since this kind of weak solutions are 4 jin and kim not unique, it is always considered with entropy condition which implies the physically meaningful solution, =-=[Lax]-=-. Asymptotic behavior of the solution v(x; s) of (2.2) is determined by two invariant variables (2.3) p = \Gamma inf x Z x \Gamma1 u 0 (y)dy; q = sup x Z 1 x u 0 (y)dy; and an N-wave (2.4) N p;q (x; s... |

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71 |
A.-Y.A well balanced scheme for numerical processing of source terms in hyperbolic equations
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Citation Context ...e the steady state solution with a better accuracy, since it is known that a scheme that ignores the steady state structure of the equations will not produce accurate steady state numerical solutions =-=[GL]-=-. Several successful numerical approaches, which take into consideration of the balance between the flux and the source term, exist in the literature [BV, BPV, CL, GHS, Go, GL, Jin, LeV1, Roe]. All th... |

64 | Balancing source term and and flux gradients in high resolution Godunov methods: The quasi-steady state wave propagation algoriyhm - LeVeque |

52 |
Nonlinear stability of shock waves for viscous conservation laws
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Citation Context ...isfied, it induces exponential growth, thus the linear instability of (1.4), just like the original system (1.1). However, the damping mechanism of the nonlinear convection (known as the N-wave decay =-=[Liu]-=-) in the Burger's equation provides a competing force with the growth term, and under suitable conditions the solution does not tend computation of roll waves 3 to infinity. For a periodic problem wit... |

42 | Some Approximate Godunov Schemes to Compute Shallow-Water Equations with Topography. Computers and Fluids - Gallouèˆt, Herard, et al. |

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8 |
On roll waves down an open inclined
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(Show Context)
Citation Context ... by a Burgers equation with a source term. Continuous roll waves in the unstable regime were constructed by Needham and Merkin using the river equations with a diffusive term in the momentum equation =-=[NM]-=-. It was realized by Whitham [Whi], and more recently explicitly formulated by Kranenburg [Kra], that one can obtain a weakly nonlinear asymptotic approximation for (1.1), which turns out to be a Burg... |

7 |
On the evolution of roll waves
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(Show Context)
Citation Context ...onstructed by Needham and Merkin using the river equations with a diffusive term in the momentum equation [NM]. It was realized by Whitham [Whi], and more recently explicitly formulated by Kranenburg =-=[Kra]-=-, that one can obtain a weakly nonlinear asymptotic approximation for (1.1), which turns out to be a Burger's equation with a source term. Specifically, if one uses the ansatz (1.3) h = h + fflh 1 (; ... |

6 |
Ocean Waves and Kindred Geophysical Phenomena
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(Show Context)
Citation Context ...ontinuous periodic traveling waves, and are known as "roll waves" [Sto]. The phenomenon of roll waves has been actually observed (and photographed!) in inclined spill ways and river beds, se=-=e Cornish [Cor]-=-. The mechanism leading to the instability was revealed by Jeffreys [Jef] in an early study of roll waves. Dressler constructed nonlinear periodic solutions that consist of piecewise smooth profiles s... |

6 |
Mathematical solution of the problem of roll-waves in inclined open channels
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(Show Context)
Citation Context ...reys [Jef] in an early study of roll waves. Dressler constructed nonlinear periodic solutions that consist of piecewise smooth profiles separated by discontinuities (shocks) using the river equations =-=[Dre]-=-. Novik [Nov] proposed to model the roll waves by a Burgers equation with a source term. Continuous roll waves in the unstable regime were constructed by Needham and Merkin using the river equations w... |

5 |
The flow of water in an inclined channel of rectangular section
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(Show Context)
Citation Context ...The phenomenon of roll waves has been actually observed (and photographed!) in inclined spill ways and river beds, see Cornish [Cor]. The mechanism leading to the instability was revealed by Jeffreys =-=[Jef]-=- in an early study of roll waves. Dressler constructed nonlinear periodic solutions that consist of piecewise smooth profiles separated by discontinuities (shocks) using the river equations [Dre]. Nov... |

5 |
Hyperbolic systems with supercharacteristic relaxations and roll waves
- Jin, Katsoulakis
(Show Context)
Citation Context ...The model (1.4) arises not only in the river equations. It is the asymptotic approximation to general hyperbolic systems with relaxations when some characteristic stability condition is violated. See =-=[JK]-=- for details. Theoretical justification of this asymptotic limit, with the addition of artificial viscosity, was made in [JK]. Model (1.4), and its asymptotic origin, the river equations (1.1), are ex... |

5 |
Asymptotic oscillations of solutions of scalar conservation laws with convexity under the action of a linear excitation
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(Show Context)
Citation Context ...l data u 0 . This implies that it suffices to consider the structure of the solution over each self-contained interval to determine the global structure. Convergence to roll waves has been studied in =-=[Lyb]-=- when initial data are periodic and have mean zero and the flux is strictly convex. We summarize some of the results we need in our context in the following lemma. Lemma 2.3. Let u(x; t) be the soluti... |

2 |
Steady state capturing methods for hyperbolic systems with source terms
- Jin
(Show Context)
Citation Context ...row algebraically and eventually becomes unbounded. Thus the success of a method for this problem depends on whether the method preserves the zero mass. We also consider another scheme, introduced in =-=[Jin]-=- for numerical capturing of the steady state of hyperbolic systems with geometrical source terms: (3.9) U n+1 j \Gamma U n j + k h ((U n j+1=2 ) 2 \Gamma (U n j \Gamma1=2 ) 2 ) = k 2 (U n j+1=2 + U n ... |

1 | Diffusive N-waves and metastability - Kim, Tzavaras |

1 |
Model description of roll-waves
- Novik
- 1971
(Show Context)
Citation Context ... an early study of roll waves. Dressler constructed nonlinear periodic solutions that consist of piecewise smooth profiles separated by discontinuities (shocks) using the river equations [Dre]. Novik =-=[Nov]-=- proposed to model the roll waves by a Burgers equation with a source term. Continuous roll waves in the unstable regime were constructed by Needham and Merkin using the river equations with a diffusi... |