On the computation of roll waves
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 463-480.

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 u t +uu x =u,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. 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.

Classification : 35L65, 65M06, 76B15
Mots-clés : roll wave, conservation laws with source term, round-off error, shock capturing methods 
@article{M2AN_2001__35_3_463_0,
     author = {Jin, Shi and Kim, Yong Jung},
     title = {On the computation of roll waves},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {463--480},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {3},
     year = {2001},
     mrnumber = {1837080},
     zbl = {1001.35084},
     language = {en},
     url = {http://archive.numdam.org/item/M2AN_2001__35_3_463_0/}
}
TY  - JOUR
AU  - Jin, Shi
AU  - Kim, Yong Jung
TI  - On the computation of roll waves
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2001
SP  - 463
EP  - 480
VL  - 35
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/item/M2AN_2001__35_3_463_0/
LA  - en
ID  - M2AN_2001__35_3_463_0
ER  - 
%0 Journal Article
%A Jin, Shi
%A Kim, Yong Jung
%T On the computation of roll waves
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2001
%P 463-480
%V 35
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/item/M2AN_2001__35_3_463_0/
%G en
%F M2AN_2001__35_3_463_0
Jin, Shi; Kim, Yong Jung. On the computation of roll waves. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 463-480. http://archive.numdam.org/item/M2AN_2001__35_3_463_0/

[1] A. Bernudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071. | Zbl

[2] R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. (to appear). | MR | Zbl

[3] A. Chinnayya and A.Y. Le Roux, A new general Riemann solver for the shallow-water equations with friction and topography. Preprint (1999).

[4] V. Cornish, Ocean waves and kindred geophysical phenomena. Cambridge University Press, London (1934).

[5] C.M. Dafermos, Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften 325, Springer-Verlag, Berlin (2000) xvi+443 pp. | MR | Zbl

[6] R.F. Dressler, Mathematical solution of the problem of roll-waves in inclined open channels. Comm. Pure Appl. Math. 2 (1949) 149-194. | Zbl

[7] T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. AIAA-2001 (to appear). | MR | Zbl

[8] J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems. Comm. Pure Appl. Math. 47 (1994) 293-306. | Zbl

[9] L. Gosse, A well-balanced flux-vector splitting scheme desinged for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135-159. | Zbl

[10] J.M. Greenberg and A.-Y. Le Roux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl

[11] 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). | Zbl

[12] H. Jeffreys, The flow of water in an inclined channel of rectangular section. Phil. Mag. 49 (1925) 793-807. | JFM

[13] S. Jin, A steady-state capturing method for hyperbolic systems with source terms. ESAIM: M2AN (to appear). | Numdam | Zbl

[14] S. Jin and M. Katsoulakis, Hyperbolic systems with supercharacteristic relaxations and roll waves. SIAM J. Appl. Math. 61 (2000) 271-292 (electronic). | Zbl

[15] Y.J. Kim and A.E. Tzavaras, Diffusive N-waves and metastability in Burgers equation. Preprint. | MR

[16] C. Kranenburg, On the evolution of roll waves. J. Fluid Mech. 245 (1992) 249-261. | Zbl

[17] 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). | MR | Zbl

[18] R. Leveque, Numerical methods for conservation laws. Lect. Math., ETH Zurich, Birkhauser (1992). | MR | Zbl

[19] R.J. Leveque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346-365. | Zbl

[20] T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Memoirs of the AMS 56 (1985). | MR | Zbl

[21] 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. | Zbl

[22] D.J. Needham and J.H. Merkin, On roll waves down an open inclined channel. Proc. Roy. Soc. Lond. A 394 (1984) 259-278. | Zbl

[23] O.B. Novik, Model description of roll-waves. J. Appl. Math. Mech. 35 (1971) 938-951.

[24] P.L. Roe, Upwind differenced schemes for hyperbolic conservation laws with source terms. Lect. Notes Math. 1270, Springer, New York (1986) 41-51. | Zbl

[25] J.J. Stoker, Water Waves. John Wiley and Sons, New York (1958). | MR | Zbl

[26] J. Whitham, Linear and nonlinear waves. Wiley, New York (1974). | MR | Zbl