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 ${u}_{t}+u{u}_{x}=u,\phantom{\rule{4pt}{0ex}}u(x,0)={u}_{0}\left(x\right),$ 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.

Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {35}, number = {3}, year = {2001}, pages = {463-480}, zbl = {1001.35084}, mrnumber = {1837080}, language = {en}, url = {http://www.numdam.org/item/M2AN_2001__35_3_463_0} }

Jin, Shi; Kim, Yong Jung. On the computation of roll waves. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 3, pp. 463-480. http://www.numdam.org/item/M2AN_2001__35_3_463_0/

[1] Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071. | Zbl 0816.76052

and ,[2] Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. (to appear). | MR 1933816 | Zbl 1017.65070

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

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

,[5] Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften 325, Springer-Verlag, Berlin (2000) xvi+443 pp. | MR 1763936 | Zbl 0940.35002

,[6] Mathematical solution of the problem of roll-waves in inclined open channels. Comm. Pure Appl. Math. 2 (1949) 149-194. | Zbl 0038.38405

,[7] Some approximate Godunov schemes to compute shallow-water equations with topography. AIAA-2001 (to appear). | MR 1966639 | Zbl 1084.76540

, and ,[8] Stability of the Kuramoto-Sivashinsky and related systems. Comm. Pure Appl. Math. 47 (1994) 293-306. | Zbl 0809.35105

,[9] 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 0963.65090

,[10] A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl 0876.65064

and ,[11] Asymptotic equations for nonlinear hyperbolic waves, in Surveys in Appl. Math. Vol. 2, J.B. Keller, G. Papanicolaou, D.W. McLaughlin, Eds. (1993). | Zbl 0856.35075

,[12] The flow of water in an inclined channel of rectangular section. Phil. Mag. 49 (1925) 793-807. | JFM 51.0668.02

,[13] A steady-state capturing method for hyperbolic systems with source terms. ESAIM: M2AN (to appear). | Numdam | Zbl 1001.35083

,[14] Hyperbolic systems with supercharacteristic relaxations and roll waves. SIAM J. Appl. Math. 61 (2000) 271-292 (electronic). | Zbl 0988.35107

and ,[15] Diffusive N-waves and metastability in Burgers equation. Preprint. | MR 1871412

and ,[16] On the evolution of roll waves. J. Fluid Mech. 245 (1992) 249-261. | Zbl 0765.76011

,[17] Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CBMS-NSF Regional Conference Series Appl. Math. 11, Philadelphia (1973). | MR 350216 | Zbl 0268.35062

,[18] Numerical methods for conservation laws. Lect. Math., ETH Zurich, Birkhauser (1992). | MR 1153252 | Zbl 0723.65067

,[19] 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 0931.76059

,[20] Nonlinear stability of shock waves for viscous conservation laws. Memoirs of the AMS 56 (1985). | MR 791863 | Zbl 0617.35058

,[21] 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 0729.35080

,[22] On roll waves down an open inclined channel. Proc. Roy. Soc. Lond. A 394 (1984) 259-278. | Zbl 0553.76013

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

,[24] Upwind differenced schemes for hyperbolic conservation laws with source terms. Lect. Notes Math. 1270, Springer, New York (1986) 41-51. | Zbl 0626.65086

,[25] Water Waves. John Wiley and Sons, New York (1958). | MR 1153414 | Zbl 0812.76002

,[26] Linear and nonlinear waves. Wiley, New York (1974). | MR 483954 | Zbl 0373.76001

,