Numerical entropy production for shallow water flows

Sudi Mungkasi, Stephen Gwyn Roberts

Abstract


A numerical scheme for the entropy of the one dimensional shallow water wave equations is presented. The scheme follows from a well-balanced finite volume method for the quantity vector having water height and momentum as its components. The local truncation error of the entropy is called the numerical entropy production, and can be used to detect the location of a shock discontinuity. We show by numerical tests that the numerical entropy production performs better in detecting such a discontinuity than two local truncation errors of the numerical quantity.

References
  • E. Audusse, F. Bouchut, M O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput., 25(6):2050--2065, 2004. doi:10.1137/S1064827503431090
  • F. Bouchut. Efficient numerical finite volume schemes for shallow water models. In V. Zeitlin, ed., Nonlinear dynamics of rotating shallow water: methods and advances, Vol. 2 of Edited series on advances in nonlinear science and complexity, pages 189--256. Elsevier, 2007. doi:10.1016/S1574-6909(06)02004-1
  • L A. Constantin, and A. Kurganov. Adaptive central-upwind schemes for hyperbolic systems of conservation laws. In F. Asakura et al., eds., Hyperbolic problems: Theory, numerics, and applications, Vol. 1, pages 95--103. Yokohama Publishers, Yokohama, 2006. http://129.81.170.14/ kurganov/Constantin-Kurganov.pdf
  • F. Golay. Numerical entropy production and error indicator for compressible flows. C. R. Mecanique, 337(4):233--ñ237, 2009. doi:10.1016/j.crme.2009.04.004
  • S. Karni, A. Kurganov, and G. Petrova. A smoothness indicator for adaptive algorithms for hyperbolic systems. J. Comput. Phys., 178(2):323--341, 2002. doi:10.1006/jcph.2002.7024
  • A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput., 23(3):707--740, 2001. doi:10.1137/S1064827500373413
  • S. Noelle, N. Pankratz, G. Puppo, and J R. Natvig. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys., 213(2):474--499, 2006. doi:10.1016/j.jcp.2005.08.019
  • G. Puppo. Numerical entropy production on shocks and smooth transitions. J. Sci. Comput., 17(1ñ--4):263--271, 2002. doi:10.1023/A:1015117118157
  • G. Puppo. Numerical entropy production for central schemes. SIAM J. Sci. Comput., 25(4):1382--ñ1415, 2003 doi:10.1137/S1064827502386712
  • E. Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica, 12:451ñ--512, 2003. doi:10.1017/S0962492902000156

Keywords


numerical entropy production; shallow water wave equations; smoothness indicator; shock detector

Full Text:

PDF BibTeX


DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3786



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.