Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry

Journal of Computational Physics

Volumn 227, Issue 6, 2008, Pages 3212-3243

  Bernetti, R ^a   Titarev, V A ^b   Toro, E F ^c  

^a UNIVERSITÀ POLITECNICA DELLE MARCHE   (Italy)

^b CRANFIELD UNIVERSITY   (United Kingdom)

^c UNIVERSITY OF TRENTO   (Italy)

Author keywords

Discontinuous bottom geometry; Exact Riemann solver; Shallow water equations; WAF method

Indexed keywords

EQUATIONS OF MOTION; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SHOCK WAVES;

ADDITIONAL EQUATIONS; DISCONTINUOUS BOTTOM GEOMETRY; EXACT RIEMANN SOLV; EXACT SOLUTION; NON-LINEAR SHALLOW WATER EQUATIONS; RIEMANN PROBLEM; RIEMANN SOLVER; SECOND ORDERS; SHALLOW WATER EQUATIONS; WAF METHOD;

GEOMETRY;

EID: 38749099435     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.11.033     Document Type: Article

References (19)

1. Alcrudo F., and Benkhaldom F. Solution to the Riemann problem of the shallow water equation with a bottom step. Comput. Fluids 30 (2001) 643-671
2. Bender C.M., and Orszag S.A. Advanced Mathematical Methods for Scientists and Engineers (1978), McGraw
3. M. Brocchini, A. Mancinelli, L. Soldini, R. Bernetti, Structure-generated macrovortices and their evolution in very shallow depths, in: Proceedings of the 28th I.C.C.E., vol. 1, Cardiff, UK, 2002, pp. 772-783.
4. Brocchini M., Bernetti R., Mancinelli A., and Albertini G. An efficient solver for near shore flows based on the waf method. Costal Eng. (2001)
5. Bukreev V.I., Gusev A.V., and Ostapenko V.V. Breakdown of a discontinuity of the free fluid surface over a bottom step in a channel. Fluid Dyn. 38 2 (2003) 889-899
6. Gallouet T., Herard J.-M., and Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 4 (2003) 479-513
7. Glaister P. Approximate Riemann solutions for the shallow water equations. J. Hydraulic Res. 23 3 (1988) 293-306
8. Godunov S.K. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sbornik 47 (1959) 357-393
9. Greenberg J.M., Leroux A.Y., Baraille R., and Noussair A. Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 5 (1997) 1980-2007
10. Hu K., Mingham C.G., and Causon D.M. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coastal Eng. 41 4 (2000) 433-465
11. P.G. LeFloch, Mai Duc Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Preprint NI03024-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 2003.
12. LeVeque R.J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365
13. Lighthill J. Waves in Fluids (1978), Cambridge University Press
14. Roe P.L. Upwind differencing schemes for hyperbolic conservation laws with source terms. In: Carasso, Raviart, and Serre (Eds). Proceedings of the First International Conference on Hyperbolic Problems (1986), Springer 41-51
15. Stoker J.J. Water Waves (1957), Interscience, New York
16. Toro E.F. A weighted average flux method for hyperbolic conservation laws. Proc. Roy. Soc. London A423 (1989) 401-418
17. Toro E.F. Riemann problems and the WAF method for solving two-dimensional shallow water equations. Phil. Trans. Roy. Soc. London (1989) 43-68
18. Toro E.F. Shock-Capturing Methods for Free-Surface Shallow Flows (2001), Wiley and Sons Ltd.
19. Vazquez-Cendon M.E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148 (1999) 497-526