Interpolation Errors in Rectangular and Diamond Characteristic Grids
Publication: Journal of Hydraulic Engineering
Volume 134, Issue 10
Abstract
Analytical predictions of numerical errors in method of characteristics analyses using time-line interpolation in a rectangular grid are obtained for (1) both time-line and space-line interpolation and (2) both rectangular and diamond grids. Amplitude and frequency errors are investigated for each of these four cases using purpose-developed polynomial transfer matrices. Both semiinfinite and finite pipes are investigated. The time-line analysis permits reach back in time and the space-line analysis permits reach out in space. A common definition is adopted for the Courant number in the four cases and it is shown why stability can be achieved in reach-out analyses with Courant numbers greater than 1. In contrast with most work on error estimation, the predicted errors are obtained analytically, not numerically. This is made possible by restricting the analysis to special, but important, cases such as liquid-filled pipes in which the waves may be assumed to propagate at constant speed. Furthermore, the development is restricted to inviscid flows, thereby enabling interpolation errors to be assessed in the absence of complicating influences of discretization errors. In contrast with the latter, it is found that interpolation errors are more sensitive to the shape of numerical grids (i.e., Courant number and rectangular versus diamond grid) than to the size of the numerical time step.
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Acknowledgments
The writers wish to express gratitude for financial assistance from the Japanese Society for the Promotion of Science (Grant-in-Aid No. UNSPECIFIED14360140) and from the UK Engineering and Physical Sciences Research Council (Grant No. UNSPECIFIEDEP/C015479/1). The writers are solely responsible for the content of the paper, which does not necessarily reflect the opinion of JSPS or the EPSRC.
References
Ahmad, Z., and Kothyari, U. C. (2001). “Time-line cubic spline interpolation scheme for solution of advection equation.” Comput. Fluids, 30, 737–752.
Chaudhry, M. H., and Hussaini, M. Y. (1985). “Second-order accurate explicit finite-difference schemes for waterhammer analysis.” J. Fluids Eng., 107, 523–529.
Ghidaoui, M. S., and Karney, B. (1994). “Equivalent differential equations in fixed-grid characteristics method.” J. Hydraul. Eng., 120(10), 1159–1175.
Ghidaoui, M. S., Karney, B., and McInnis, D. A. (1998). “Energy estimates for discretization errors in water hammer problems.” J. Hydraul. Eng., 124(4), 384–393.
Goldberg, D., and Wylie, E. B. (1983). “Characteristics method using time-line interpolations.” J. Hydraul. Eng., 109(5), 670–683.
Karney, B. (1990). “Energy relations in transient closed-conduit flow.” J. Hydraul. Eng., 116(10), 1180–1196.
Karney, B. W., and Ghidaoui, M. S. (1997). “Flexible discretization algorithm for fixed-grid MOC in pipelines.” J. Hydraul. Eng., 123(11), 1004–1011.
Lai, C. (1989). “Comprehensive method of characteristics models for flow simulation.” J. Hydraul. Eng., 114(9), 1074–1095.
Maudsley, D. (1984). “Errors in the simulation of pressure transients in a hydraulic system.” Trans. Inst. Meas. Control (London), 6(1), 7–12.
Murray, S. J. (1992). “Transient multi-regime pipe flow—A case study.” Proc., Int. Conf. on Unsteady Flow and Fluid Transients, Durham, U.K., HR Wallingford, 121–136.
Shimada, M. (1994). “Theoretical analysis on numerical error of MOC-based time-line method predicting pipe flows and grid generation of combined time-line method.” Trans. Jpn. Soc. Irrig. Drain. Reclam. Eng., 173(10), 79–84 (in Japanese with an abstract in English).
Shimada, M., Brown, J. M. B., Leslie, D., and Vardy, A. E. (2006). “Time-line interpolation errors in pipe networks.” J. Hydraul. Eng., 132, 294–306.
Shimada, M., Brown, J. M. B., and Vardy, A. E. (2007). “Estimating friction errors in MOC analyses of unsteady pipe flows.” Comput. Fluids, 32, 1235–1246.
Shimada, M., and Vardy, A. E. (2000). “Towards automatic error limitation in MOC analyses.” Proc., 8th Int. Conf. on Pressure Surges, The Hague, The Netherlands, BHR Group, 183–204.
Sibetheros, I. A., Holley, E. R., and Branski, J. M. (1991). “Spline interpolations for waterhammer analysis.” J. Hydraul. Eng., 117(10), 1332–1351.
Sowlati, T., and Karney, B. W. (2001). “Analysis of interpolation strategies in fixed-grid method of characteristics solution in open channels.” Proc., 29th IAHR Congress, Beijing, IAHR, 927–932.
Streeter, V. L. (1972). “Unsteady flow calculations by numerical methods.” J. Basic Eng., 94, 457–466.
Tsai, T.-L., Chiang, S.-W., and Yang, J.-C. (2004). “Characteristics method with cubic-spline interpolation for open channel flow computation.” Int. J. Numer. Methods Fluids, 46, 663–683.
Tsai, T.-L., Chiang, S.-W., and Yang, J.-C. (2006). “Examination of characteristics method with cubic interpolation for advection–diffusion equation.” Comput. Fluids, 35, 1217–1227.
Vitkovsky, J., Lambert, M., Simpson, A., and Bergant, A. (2000). “Advances in unsteady friction modelling in transient pipe flow.” Proc., 8th Int. Conf. on Pressure Surges, The Hague, The Netherlands, BHR Group, 471–482.
Wiggert, D. C., and Sundquist, M. J. (1977). “Fixed-grid characteristics for pipeline transients.” J. Hydr. Div., 103(12), 1403–1416.
Wylie, E. B. (1980). “Inaccuracies in the characteristics method.” Proc., Speciality Conf. on Computer and Physical Modelling in Hydraulic Engineering, ASCE, Reston, Va., 165–176.
Wylie, E. B., and Streeter, V. L. (1993). Fluid transients in systems, Prentice-Hall, Upper Saddle River, N.J.
Yang, J.-C., and Hsu, E.-L. (1990). “Time-line interpolation for solution of the dispersion equation.” J. Hydraul. Res., 28(4), 503–520.
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Published In
Journal of Hydraulic Engineering
Volume 134 • Issue 10 • October 2008
Pages: 1480 - 1490
Copyright
© 2008 ASCE.
History
Received: Jun 20, 2006
Accepted: Mar 13, 2008
Published online: Oct 1, 2008
Published in print: Oct 2008
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