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http://dx.doi.org/10.12652/Ksce.2012.32.1B.021

An Application of the HLLL Approximate Riemann Solver to the Shallow Water Equations  

Hwang, Seung-Yong (한국건설기술연구원 수자원.환경연구본부 하천해안연구실)
Lee, Sam Hee (한국건설기술연구원 수자원.환경연구본부 하천해안연구실)
Publication Information
KSCE Journal of Civil and Environmental Engineering Research / v.32, no.1B, 2012 , pp. 21-27 More about this Journal
Abstract
The HLLL scheme, proposed by T. Linde, determines all the wave speeds from the initial states because the middle wave is evaluated by the introduction of a generalized entropy function. The scheme is considered a genuine successor to the original HLL scheme because it is completely separated form the Roe's linearization scheme unlike the HLLE scheme and does not rely on the exact solution unlike the HLLC scheme. In this study, a numerical model was configured by the HLLL scheme with the total energy as a generalized entropy function to solve governing equations, which are the one-dimensional shallow water equations without source terms and with an additional conserved variable relating a concentration. Despite the limitations of the first order solutions, results to three cases with the exact solutions were generally accurate. The HLLL scheme appeared to be superior in comparison with the other HLL-type schemes. In particular, the scheme gave fairly accurate results in capturing the front of wetting and drying. However, it revealed shortcomings of more time-consuming calculations compared to the other schemes.
Keywords
shallow water equations; approximate Riemann solver; HLL scheme; HLLL scheme;
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  • Reference
1 Einfeldt, B. (1988) On Godunov-type methods for gas dynamics. SIAM Journal on Numerical Analysis, Vol. 25, pp. 294-318.   DOI   ScienceOn
2 Fjordholm, U.S., Mishra, S., and Tadmor, E. (2011) Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. Journal of Computational Physics, Vol. 230, pp. 5587-5609.   DOI   ScienceOn
3 Fraccarollo, L. and Toro, E.F. (1995) Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. Journal of Hydraulic Research, Vol. 33, pp. 843-864.   DOI   ScienceOn
4 George, D.L. (2006) Finite volume methods and adaptive refinement for Tsunami propagation and inundation. Ph.D. dissertation, University of Washington.
5 Harten, A., Lax, P.D., and van Leer, B. (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, Vol. 25, pp. 35-61.   DOI   ScienceOn
6 LeVeque, R.J. (2002) Finite volume method for hyperbolic problems. Cambridge University Press.
7 Linde, T. (2002) A practical, general-purpose, two-state HLL Riemann solver for hyperbolic conservation laws. International Journal for Numerical Methods in Fluids, Vol. 40, pp. 391-402.   DOI
8 Roe, P.L. (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, Vol. 43, pp. 357-372.   DOI   ScienceOn
9 Suzuki, Y., Khieu, L., and van Leer, B. (2009) CFD by first order PDEs. Continuum Mechanics and Thermodynamics, Vol. 21, pp. 445-465.   DOI   ScienceOn
10 Tadmor, E. (1984) Skew-selfadjoint form for systems of conservation laws. Journal of Mathematical Analysis and Applications, Vol. 103, pp. 428-442.   DOI   ScienceOn
11 Toro, E.F. (2001) Shock-capturing methods for free-surface shallow flows. John Wiley & Sons.
12 Toro, E.F., Spruce, M., and Speares W. (1994) Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, Vol. 4, pp. 25-34.   DOI   ScienceOn
13 Trangenstein, J.A. (2009) Numerical solution of hyperbolic partial differential equations. Cambridge University Press.
14 van Leer, B. (2006) Upwind and high-resolution method for compressible flow: from donor cell to residual-distribution schemes. Communications in Computational Physics, Vol. 1, pp. 192- 206.
15 van Leer, B. (2009) The development of numerical fluid mechanics and aerodynamics since the 1960s: US and Canada. 100 volumes of 'notes on numerical fluid mechanics', E. H. Hirschel and E. Krause, eds., Springer-Verlag, pp. 159-185.
16 Wackers, J. and Koren, B. (2004) Five-equation model for compressible two-fluid flow. Report MAS-E0414, CWI.
17 Weiyan, T. (1992) Shallow water hydrodynamics. Elsevier Science Publishers.