Equivalent Differential Equations in Fixed‐Grid Characteristics Method
Publication: Journal of Hydraulic Engineering
Volume 120, Issue 10
Abstract
Various numerical techniques, such as wave‐speed adjustment or interpolation, are generally required in order to apply the fixed‐grid method of characteristics to multipipe systems or to systems with variable wave speed. However, these techniques introduce into the solution unwanted side effects such as numerical attenuation and dispersion. The present paper develops the concept of an equivalent hyperbolic differential equation to study how discretization errors arise in pipeline applications for the most common interpolation techniques. In particular, it is shown that space‐line interpolation and the Holly‐Preissmann scheme are equivalent to a wave‐diffusion model with an adjusted wave speed, but that the latter method has additional source and sink terms. Further, time‐line interpolation is shown to be equivalent to a superposition of two waves with different wave speeds. In general, the equivalent hyperbolic differential equation concept evaluates the consistency of the numerical scheme, provides a mathematical description of the numerical dissipation and dispersion, gives an independent way of determining the Courant condition, allows the comparison of alternative approaches, finds the wave path, and explains why higher‐order methods should usually be avoided.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Chaudhry, M. H., and Hussaini, M. Y. (1985). “Second‐order accurate explicit finitedifference schemes for waterhammer analysis.” J. Fluid Eng., 107(4), 523–529.
2.
Dammuller, D. C., Bhallamudi, S. M., and Chaudhry, M. H. (1989). “Modelling unsteady flow in curved channel.” J. Hydr. Engrg., ASCE, 115(11), 1479–1495.
3.
Ghidaoui, M. S., and Karney, B. W. (1992). “Time discretization in fixed grid method of characteristics.” Proc., Annu. Conf. of the CSCE, Canadian Society of Civil Engineers (CSCE), Quebec City, Canada, 367–376.
4.
Goldberg, D. E., and Wylie, E. B. (1983). “Characteristics method using time‐line interpolations.” J. Hydr. Engrg., ASCE, 109(5), 670–683.
5.
Guenther, R. B., and Lee, J. W. (1988). Partial differential equations of mathematical physics and integral equations. Prentice‐Hall, Englewood Cliffs, N.J.
6.
Holly, M., and Preissmann, A. (1977). “Accurate calculation of transport in two dimensions.” J. Hydr. Engrg., ASCE, 103(11), 1259–1277.
7.
Katopodes, N. D. (1984). “Fourier analysis of dissipative FEM channel flow model.” J. Hydr. Engrg., ASCE, 110(7), 927–944.
8.
Lai, C. (1989). “Comprehensive method of characteristics models for flow simulation.” J. Hydr. Engrg., ASCE, 114(9), 1074–1095.
9.
Lavooij, C. S. W., and Tijsseling, A. S. (1991). “Fluid‐structure interation in liquidfilled piping systems.” J. Fluids and Struct., 5, 573–595.
10.
Leonard, B. P. (1979). “A stable and accurate convective modelling procedure based on quadratic upstream interpolation.” Comp. Methods in Appl. Mech. and Engrg., 19, 59–98.
11.
Noye, B. J. (1991). “Some three‐level finite difference methods for simulating advection in fluids.” Comp. & Fluids, 19(1), 119–140.
12.
O'Brian, G. G., Hyman, M. A., and Kaplan, S. (1951). “A study of the numerical solution of partial differential equations.” J. Math. Phys., 29(4), 223–251.
13.
Samuels, G. P., and Skeels, P. C. (1990). “Stability limits for Preissmann's scheme.” J. Hydr. Engrg., ASCE, 116(8), 997–1011.
14.
Schohl, G. A., and Holly, F. M. (1991). “Cubic‐spline interpolation in Lagrangian advection computation.” J. Hydr. Engrg., ASCE, 117(2), 248–253.
15.
Sibetheros, I. A., Holley, E. R., and Branski, J. M. (1991). “Spline interpolations for water hammer analysis.” J. Hydr. Engrg., ASCE, 117(10), 1332–1349.
16.
Sivaloganathan, K. (1978). “Flood routing by characteristic methods.” J. Hydr. Div., ASCE, 107(7), 1075–1091.
17.
Wiggert, D. C., and Sundquist, M. J. (1977). “Fixed‐grid characteristics for pipeline transients.” J. Hydr. Div., ASCE, 103(12), 1403–1415.
18.
Wylie, E. B. (1980). “Inaccuracies in the characteristics method.” Proc., Spec. Conf. on Comp. and Physical Modelling in Hydr. Engrg., ASCE, New York, N.Y., 165—176.
19.
Wylie, E. B., and Streeter, V. L. (1982). Fluid transients. Feb Press, Ann Arbor, Mich.
20.
Yanenko, N. N., Fedotova, Z. I., Tusheva, L. A., and Shokin, Y. I. (1983). “Classification of difference schemes of gas dynamics by the method of differential approximation—I.” Comp. and Fluids, 11(3), 187–206.
21.
Yang, J. C., and Hsu, E. L. (1990). “Time‐line interpolation for solution of the dispersion equation.” J. Hydr. Res., 28(4), 503–523.
Information & Authors
Information
Published In
Copyright
Copyright © 1994 American Society of Civil Engineers.
History
Received: Feb 19, 1993
Published online: Oct 1, 1994
Published in print: Oct 1994
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.