On the Computation of Roll Waves

Shi Jin; Yong Jung Kim

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 3, page 463-480
  • ISSN: 0764-583X

Abstract

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The phenomenon of roll waves occurs in a uniform open-channel flow down an 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 ut + uux = u,u(x,0) = u0(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. In this paper, we first study the analytic behavior of the solution to the above model. We then discuss the numerical difficulty, and introduce a numerical method that predicts precisely the evolution and steady state of its solution. Various numerical experiments are performed to illustrate the numerical difficulty and the effectiveness of the proposed numerical method.

How to cite

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Jin, Shi, and Kim, Yong Jung. "On the Computation of Roll Waves." ESAIM: Mathematical Modelling and Numerical Analysis 35.3 (2010): 463-480. <http://eudml.org/doc/197563>.

@article{Jin2010,
abstract = { The phenomenon of roll waves occurs in a uniform open-channel flow down an 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 ut + uux = u,u(x,0) = u0(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. In this paper, we first study the analytic behavior of the solution to the above model. We then discuss the numerical difficulty, and introduce a numerical method that predicts precisely the evolution and steady state of its solution. Various numerical experiments are performed to illustrate the numerical difficulty and the effectiveness of the proposed numerical method. },
author = {Jin, Shi, Kim, Yong Jung},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Roll wave; conservation laws with source term; round-off error; shock capturing methods.; conservation laws with source term; shock capturing methods},
language = {eng},
month = {3},
number = {3},
pages = {463-480},
publisher = {EDP Sciences},
title = {On the Computation of Roll Waves},
url = {http://eudml.org/doc/197563},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Jin, Shi
AU - Kim, Yong Jung
TI - On the Computation of Roll Waves
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 3
SP - 463
EP - 480
AB - The phenomenon of roll waves occurs in a uniform open-channel flow down an 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 ut + uux = u,u(x,0) = u0(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. In this paper, we first study the analytic behavior of the solution to the above model. We then discuss the numerical difficulty, and introduce a numerical method that predicts precisely the evolution and steady state of its solution. Various numerical experiments are performed to illustrate the numerical difficulty and the effectiveness of the proposed numerical method.
LA - eng
KW - Roll wave; conservation laws with source term; round-off error; shock capturing methods.; conservation laws with source term; shock capturing methods
UR - http://eudml.org/doc/197563
ER -

References

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