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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. ([email protected].)
Yong Jung Kim
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455-0436, USA. ([email protected])
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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.
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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References

Bernudez, A. and Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071.
R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. (to appear).
A. Chinnayya and A.Y. Le Roux, A new general Riemann solver for the shallow-water equations with friction and topography. Preprint (1999).
V. Cornish, Ocean waves and kindred geophysical phenomena. Cambridge University Press, London (1934).
C.M. Dafermos, Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften 325, Springer-Verlag, Berlin (2000) xvi+443 pp.
Dressler, R.F., Mathematical solution of the problem of roll-waves in inclined open channels. Comm. Pure Appl. Math. 2 (1949) 149-194. CrossRef
T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. AIAA-2001 (to appear).
Goodman, J., Stability of the Kuramoto-Sivashinsky and related systems. Comm. Pure Appl. Math. 47 (1994) 293-306. CrossRef
Gosse, L., A well-balanced flux-vector splitting scheme desinged for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135-159. CrossRef
Greenberg, J.M. and Le Roux, A.-Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. CrossRef
J.K. Hunter, Asymptotic equations for nonlinear hyperbolic waves, in Surveys in Appl. Math. Vol. 2, J.B. Keller, G. Papanicolaou, D.W. McLaughlin, Eds. (1993).
Jeffreys, H., The flow of water in an inclined channel of rectangular section. Phil. Mag. 49 (1925) 793-807. CrossRef
S. Jin, A steady-state capturing method for hyperbolic systems with source terms. ESAIM: M2AN (to appear).
S. Jin and M. Katsoulakis, Hyperbolic systems with supercharacteristic relaxations and roll waves. SIAM J. Appl. Math. 61 (2000) 271-292 (electronic).
Y.J. Kim and A.E. Tzavaras, Diffusive N-waves and metastability in Burgers equation. Preprint.
Kranenburg, C., On the evolution of roll waves. J. Fluid Mech. 245 (1992) 249-261. CrossRef
P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CBMS-NSF Regional Conference Series Appl. Math. 11, Philadelphia (1973).
R. LeVeque, Numerical methods for conservation laws. Lect. Math., ETH Zurich, Birkhauser (1992).
LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346-365. CrossRef
T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Memoirs of the AMS 56 (1985).
A.N. Lyberopoulos, Asymptotic oscillations of solutions of scalar conservation laws with convexity under the action of a linear excitation. Quart. Appl. Math. XLVIII (1990) 755-765.
Needham, D.J. and Merkin, J.H., On roll waves down an open inclined channel. Proc. Roy. Soc. Lond. A 394 (1984) 259-278. CrossRef
Novik, O.B., Model description of roll-waves. J. Appl. Math. Mech. 35 (1971) 938-951. CrossRef
P.L. Roe, Upwind differenced schemes for hyperbolic conservation laws with source terms. Lect. Notes Math. 1270, Springer, New York (1986) 41-51.
J.J. Stoker, Water Waves. John Wiley and Sons, New York (1958).
J. Whitham, Linear and nonlinear waves. Wiley, New York (1974).