Wavelet-Galerkin Solution to the Water Hammer Equations
Publication: Journal of Hydraulic Engineering
Volume 135, Issue 4
Abstract
In this paper, a wavelet-Galerkin method is utilized to solve the hyperbolic partial differential equations describing transient flow in a simple pipeline. Two wavelets (Haar and Daubechies) are utilized as bases for the Galerkin scheme. The governing equations are solved for the expansion coefficients, which are then used to reconstruct the signal at the downstream end of the pipeline; the computed results are in an excellent agreement with those calculated by using the method of characteristics including laminar or linearized turbulent friction terms. Most importantly, the wavelet-Galerkin approach allows the transient flow equations to be solved directly for the expansion coefficients at a certain level of resolution. This can be used to form the wavelet multiresolution framework that can be utilized for further analysis, such as feature extraction and signal identification.
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References
Anderson, E., et al. (1999). LAPACK user’s guide, 3rd Ed., SIAM, Philadelphia.
Boggess, A., and Narcowich, F. J. (2001). A first course in wavelets with Fourier analysis, Prentice-Hall, Upper Saddle River, N.J.
Chen, M., Hwang, C., and Shih, Y. (1996). “The computation of wavelet-Galerkin approximation on a bounded interval.” Int. J. Numer. Methods Eng., 39, 2921–2944.
Daubechies, I. (1988). “Orthonormal bases of compactly supported wavelets.” Commun. Pure Appl. Math., 41, 909–996.
Ferrante, M., and Brunone, B. (2003). “Pipe system diagnosis and leak detection by unsteady-state tests. II: Wavelet analysis.” Adv. Water Resour., 26, 107–116.
Glowinski, R., Lawton, W., Ravachol, M., and Tenenbaum, E. (1990). “Wavelet solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension.” Proc. 9th Int. Conf. on Numerical Methods in Applied Sciences and Engineering, SIAM, Philadelphia.
Latto, A., Resnikoff, H., and Tenenbaum, E. (1992). “The evaluation of connection coefficients of compactly supported wavelets.” Aware Technical Rep. No. AD910708; and, Proc. of the French–USA Workshop on Wavelets and Turbulence, Princeton University, Y. Maday, ed., Springer, New York.
Latto, A., and Tenenbaum, E. (1990). “Les ondelettes à support compact et la solution numérique de l’équation de Burgers.” C. R. Acad. Sci. III, 311, 903–909.
Mallat, S. (1989). “A theory for multiresolution signal decomposition: The wavelet transform.” Trans. Am. Math. Soc., 315, 68–87.
Qian, S., and Weiss, J. (1993). “Wavelets and the numerical solution of partial differential equations.” J. Comput. Phys., 106, 155–175.
Restrepo, J., and Leaf, G. (1995). “Wavelet-Galerkin Discretization of Hyperbolic Equations.” J. Comput. Phys., 122, 118–128.
Romine, C. H., and Peyton, B. W. (1997). “Computing connection coefficients of compactly supported wavelets on bounded intervals.” U.S. Department of Energy, DE-AC05–96OR, Washington, D.C.
Stoianov, I., Karney, B., Covas, D., Maksimovic, C., and Graham, N. (2002). “Wavelet processing of transient signals for pipeline leak location and quantification.” Proc., 1st Annual Environmental & Water Resources Systems Analysis Symp. in Conjunction with ASCE Environmental & Water Resources Institute Annual Conf., ASCE, New York.
Weiss, J. (1992). “Wavelets and the study of two-dimensional turbulence.” Aware Technical Rep. No. AD910628; and, Proc. French–USA Workshop on Wavelets and Turbulence, Princeton University, Y. Maday, ed., Springer, New York.
Wood, D. J., Dorsch, R. G., and Lightner, C. (1966). “Wave-plan analysis of unsteady flow in closed conduits.” J. Hydr. Div., 92(2), 83.
Zhang, T., Tian, Y. C., Tade, M. O., and Uto, O. J. (2007). “Comments on the computation of wavelet-Galerkin approximation on a bounded interval.” Int. J. Numer. Methods Eng., 72, 244–251.
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Information
Published In
Journal of Hydraulic Engineering
Volume 135 • Issue 4 • April 2009
Pages: 283 - 295
Copyright
© 2009 ASCE.
History
Received: Oct 2, 2006
Accepted: Oct 8, 2008
Published online: Apr 1, 2009
Published in print: Apr 2009
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