On the computation of roll waves

Shi Jin; Yong Jung Kim

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

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

Abstract

top
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 , 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.

How to cite

top

Jin, Shi, and Kim, Yong Jung. "On the computation of roll waves." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.3 (2001): 463-480. <http://eudml.org/doc/194058>.

@article{Jin2001,
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 $u_t+uu_x=u,\ u(x,0)=u_0(x),$ 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.},
author = {Jin, Shi, Kim, Yong Jung},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {roll wave; conservation laws with source term; round-off error; shock capturing methods},
language = {eng},
number = {3},
pages = {463-480},
publisher = {EDP-Sciences},
title = {On the computation of roll waves},
url = {http://eudml.org/doc/194058},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Jin, Shi
AU - Kim, Yong Jung
TI - On the computation of roll waves
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
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 $u_t+uu_x=u,\ u(x,0)=u_0(x),$ 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.
LA - eng
KW - roll wave; conservation laws with source term; round-off error; shock capturing methods
UR - http://eudml.org/doc/194058
ER -

References

top
  1. [1] A. Bernudez and M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. Zbl0816.76052
  2. [2] R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. (to appear). Zbl1017.65070MR1933816
  3. [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. [4] V. Cornish, Ocean waves and kindred geophysical phenomena. Cambridge University Press, London (1934). 
  5. [5] C.M. Dafermos, Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften 325, Springer-Verlag, Berlin (2000) xvi+443 pp. Zbl0940.35002MR1763936
  6. [6] R.F. Dressler, Mathematical solution of the problem of roll-waves in inclined open channels. Comm. Pure Appl. Math. 2 (1949) 149–194. Zbl0038.38405
  7. [7] T. Gallouët, J.-M. Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. AIAA-2001 (to appear). Zbl1084.76540MR1966639
  8. [8] J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems. Comm. Pure Appl. Math. 47 (1994) 293–306. Zbl0809.35105
  9. [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. Zbl0963.65090
  10. [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. Zbl0876.65064
  11. [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). Zbl0856.35075
  12. [12] H. Jeffreys, The flow of water in an inclined channel of rectangular section. Phil. Mag. 49 (1925) 793–807. Zbl51.0668.02JFM51.0668.02
  13. [13] S. Jin, A steady-state capturing method for hyperbolic systems with source terms. ESAIM: M2AN (to appear). Zbl1001.35083
  14. [14] S. Jin and M. Katsoulakis, Hyperbolic systems with supercharacteristic relaxations and roll waves. SIAM J. Appl. Math. 61 (2000) 271–292 (electronic). Zbl0988.35107
  15. [15] Y.J. Kim and A.E. Tzavaras, Diffusive N-waves and metastability in Burgers equation. Preprint. Zbl1077.35092MR1871412
  16. [16] C. Kranenburg, On the evolution of roll waves. J. Fluid Mech. 245 (1992) 249–261. Zbl0765.76011
  17. [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). Zbl0268.35062MR350216
  18. [18] R. LeVeque, Numerical methods for conservation laws. Lect. Math., ETH Zurich, Birkhauser (1992). Zbl0723.65067MR1153252
  19. [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. Zbl0931.76059
  20. [20] T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws. Memoirs of the AMS 56 (1985). Zbl0617.35058MR791863
  21. [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. Zbl0729.35080
  22. [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. Zbl0553.76013
  23. [23] O.B. Novik, Model description of roll-waves. J. Appl. Math. Mech. 35 (1971) 938–951. 
  24. [24] P.L. Roe, Upwind differenced schemes for hyperbolic conservation laws with source terms. Lect. Notes Math. 1270, Springer, New York (1986) 41–51. Zbl0626.65086
  25. [25] J.J. Stoker, Water Waves. John Wiley and Sons, New York (1958). Zbl0812.76002MR1153414
  26. [26] J. Whitham, Linear and nonlinear waves. Wiley, New York (1974). Zbl0373.76001MR483954

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.