A new analytical solution for testing debris avalanche numerical models

Sudi Mungkasi, Stephen Gwyn Roberts


An analytical solution to a debris avalanche problem in the one dimensional topography-linked coordinate system was found by Mangeney, Heinrich, and Roche [Pure Appl. Geophys., 157:1081--1096, 2000]. We derive an analytical solution to a debris avalanche problem in the standard Cartesian coordinate system. Characteristics and a transformation technique obtain the analytical solution. This analytical solution is used to test finite volume methods with reconstruction of the conserved quantities based on: either stage, height, and velocity; or stage, height, and momentum. Numerical tests show that the finite volume method with reconstruction based on stage, height, and momentum is slightly more accurate in solving the debris avalanche problem.

  • C. Ancey, R. M. Iverson, M. Rentschler, and R. P. Denlinger. An exact solution for ideal dam-break floods on steep slopes. Water Resour. Res., 44(1):W01430 pages 1--10, 2008. doi:10.1029/2007WR006353
  • 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
  • R. F. Dressler. Unsteady non-linear waves in sloping channels. Proc. Royal Soc. London, Ser. A, 247(1249):186--198, 1958. doi:10.1098/rspa.1958.0177
  • 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
  • A. Mangeney, P. Heinrich, and R. Roche. Analytical solution for testing debris avalanche numerical models. Pure Appl. Geophys., 157(6ñ--8):1081ñ--1096, 2000. doi:10.1007/s000240050018
  • S. Mungkasi, and S. G. Roberts. On the best quantity reconstructions for a well balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface. ANZIAM J., 51(EMAC2009):C48ñ--C65, 2010. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/2576
  • 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
  • A. Ritter. Die fortpflanzung der wasserwellen. Zeitschrift des Vereines Deutscher Ingenieure, 36(33):947ñ--954, 1892.
  • J. J. Stoker. Water Waves: The Mathematical Theory with Application. Interscience Publishers, New York, 1957.


debris avalanche; dam break; Saint-Venant approach; finite volume method

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DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3785

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