Strong Stability Preserving Runge-Kutta Methods Applied to Water Hammer Problem
DOI:
https://doi.org/10.5540/tcam.2022.023.01.00063Keywords:
Method of Characteristic, Method of Lines, Strong Stability Preserving Methods, Water HammerAbstract
The characteristic method of lines is the most used numerical method applied to the water hammer problem. It transforms a system of partial differential equations involving the independent variables time and space in two ordinary differential equations along the characteristics curves and then solve it numerically. This approach, although showing great stability and quick execution time, creates ∆x-∆t dependency to properly model the phenomenon. In this article we test a different approach, using the method of lines in the usual form, without the characteristics curves and then applying strong stability preserving Runge-Kutta Methods aiming to get stability with greater ∆t.
References
S. Gottlieb, D. Ketcheson, and C.-W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific, 2011.
S. Gottlieb and C.-W. Shu, “Total variation diminishing runge-kutta schemes,” ICASE Report, 06 1996.
M. H. Chaudry, Applied Hydraulic Transients. New York: Litton Educacional
Publishing, 1979.
J. A. Fox, Hydraulic Analysis of Unsteady Flow in Pipe Networks. The Macmillian Press LTD, 1977.
V. L. Streeter, “Unsteady flow calculations by numerical methods,” Journal of Basic Engineering, pp. 457–465, 06 1972.
C. C. Gumier, Aplicação de Modelo Matemático de Simulação-Otimização na Gestão de Perda de água em Sistemas de Abastecimento. PhD thesis, Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas, Campinas, SP, 2005.
D. F. G. Santiago, D. H. Paula, F. J. da Silva, D. B. Santos, E. da Conceição Silva, A. F. Antunis, N. da Silva, and D. R. Trindade, “Modelagem matemática da carga hidráulica e vazão em condutos,” ForScience, vol. 7, 06 2019.
R. Wichowski, “Hydraulic transients analysis in pipe networks by the method of characteristics (moc),” Archives of Hydro-Engineering and Environmental Mechanics, vol. 53, no. 3, pp. 267–291, 2006.
M. B. Abbott, An Introduction to The Method of Characteristics. Thames and Hudson London, 1966.
S. Gottlieb, C. Shu, and E. Tadmor, “Strong stability preserving high order
time discretization methods,” SIAM Review, vol. 43, no. 1, pp. 89–112, 2001.
D. Ketcheson, S. Gottlieb, and C. Macdonald, “Strong stability preserving two step runge-kutta methods,” SIAM Journal on Numerical Analysis, vol. 49, 06 2011.
S. J. Ruuth, “Global optimization of explicit strong-stability-preserving runge-kutta methods,” Mathematics of Computation, vol. 75, no. 253, pp. 183–207, 2005.
C. Shu, “Total-variation-diminishing time discretizations,” SIAM J. Sci.
Statist. Comput, vol. 9, no. 6, pp. 1073–1084, 1988.
R. J. Spiteri and S. J. Ruuth, “A new class of optimal high-order strong-
stability-preserving time-stepping schemes„” SIAM J. Numer. Anal, vol. 40,
no. 2, pp. 469–491, 2002.
C. Shu, Differential Quadrature. Springer-Verlag London Limited, 1962.
L. Isherwood, Z. Grant, and S. Gottlieb, “Downwinding for preserving strong
stability in explicit integrating factor runge–kutta methods,” 10 2018.
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