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On the Computation of Roll Waves

Published online by Cambridge University Press:  15 April 2002

Shi Jin and
Yong Jung Kim
Shi Jin
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. (jin@math.wisc.edu.)
Yong Jung Kim
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455-0436, USA. (yjkim@ima.umn.edu)
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Abstract

The phenomenon of roll waves occurs in a uniform open-channelflow 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 waterequations. The main difficulty associated with the numerical approximation ofthis problem is its linear instability. Numerical round-off errorcan easily overtake the numerical solution and yields false roll wavesolution at the steady state.In this paper, we first study the analytic behavior of the solution to the abovemodel. We then discuss the numerical difficulty, and introduce a numericalmethod that predicts precisely the evolution and steady state of itssolution. Various numerical experiments are performed to illustratethe numerical difficulty and the effectiveness of the proposed numericalmethod.

Keywords

Roll waveconservation lawswith source termround-off errorshock capturing methods.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 3 , May 2001, pp. 463 - 480
Copyright
© EDP Sciences, SMAI, 2001

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