Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T04:56:15.103Z Has data issue: false hasContentIssue false
  • English
  • Français

Investigation of Effect of Endpoint Constraint on Time-Line Cubic Spline Interpolation

Published online by Cambridge University Press:  05 May 2011

T.-L. Tsai  and
J.-Y. Chen
T.-L. Tsai*
Affiliation:
Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi, Taiwan 60004, R.O.C.
J.-Y. Chen*
Affiliation:
Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi, Taiwan 60004, R.O.C.
*
*Assistant Professor
**Graduate student
Get access

Abstract

In this study, the effects of various endpoint constraints, including the first derivative, natural, quadratic, and not-a-knot endpoint constraints, on the time-line cubic spline interpolation are examined by solving the advection-diffusion equation with constant and variable velocities, and the viscous Burgers equation. The natural endpoint constraint could produce large numerical diffusion. The quadratic endpoint constraint can decrease the numerical diffusion from the natural endpoint constraint, but it could trigger large numerical oscillation. The first derivative endpoint constraint with higher-order central difference approximation has better simulated results than the natural and quadratic endpoint constraints. It can significantly reduce not only the numerical diffusion produced by the natural endpoint constraint, but also the numerical oscillation caused by the quadratic endpoint constraint. The applicability of the time-line cubic spline interpolation together with the not-a-knot endpoint constraint is limited, since the computational instability is caused while the Courant number is greater than unity. Therefore, as far as accuracy and applicability are concerned, the first derivative endpoint constraint with higher-order central difference approximation rather than the commonly used natural endpoint constraint should be a better choice for the time-line cubic spline interpolation.

Keywords

Characteristics methodCubic spline interpolationEndpoint constraintTime-line interpolation.
Type
Articles
Information
Journal of Mechanics , Volume 25 , Issue 2 , June 2009 , pp. 151 - 160
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Chang, W. Y., Lee, L. C., Lien, J. S. and Lai, J. S., “Simulations of Dam-Break Flows Using Free Surface Capturing Method,” Journal of Mechanics, 24, pp. 391–403 (2008).Google Scholar
2.Guo, W. D., Lai, J. S. and Lin, G. F., “Hybrid Flux-Spliting Finite-Volume Scheme for the Shallow Water Flow Simulation with Source Terms,” Journal of Mechanics, 23, pp. 399–414 (2007).Google Scholar
3.Holly, F. M. Jr. and Preissmann, A., “Accurate Calculation of Transport in Two Dimensions,” J. Hydr. Div., 103, pp. 1259–1277 (1977).Google Scholar
4.Yang, J. C. and Wang, J. Y., “Numerical Solution of Dispersion in One Dimension,” J. Chin. Inst. Eng., 11, pp. 379–383 (1988).CrossRefGoogle Scholar
5.Yang, J. C. and Hsu, E. L., “On the Use of the Reach-Back Characteristics Method for Calculation of Dispersion,” Int. J. Numer. Methods in Fluids, 12, pp. 225–235 (1991).CrossRefGoogle Scholar
6.Yang, J. C. and Hsu, E. L., “Time-Line Interpolation for Solution of the Dispersion Equation,” J. Hydraul. Res., 28, pp. 503–520 (1990).Google Scholar
7.Schohl, G. A. and Holly, F. M. Jr., “Cubic-Spline Interpolation in Lagrangian Advection Computation,” J. Hydr. Engrg., 117, pp. 248–253 (1991).Google Scholar
8.Karpik, S. R. and Crockett, S. R., “Semi-Lagrangian Algorithm for Two-Dimensional Advection-Diffusion Equation on Curvilinear Coordinate Meshes,” J. Hydr. Engrg., 123, pp. 389–401 (1997).Google Scholar
9.Stefanovic, D. L. and Stefan, H. G., “Accurate Two-Dimensional Simulation of Advective -Diffusive -Reactive Transport,” J. Hydr. Engrg., 127, pp. 728–737 (2001).Google Scholar
10.Ahmad, Z. and Kothyari, U. C., “Time-Line Cubic Spline Interpolation Scheme for Solution of Advection Equation,” Computers and Fluids, 30, pp. 737–752 (2001).CrossRefGoogle Scholar
11.Tsai, T. L., Yang, J. C. and Huang, L. H., “Characteristics Method Using Cubic-Spline Interpolation for Advection-Diffusion Equation,” J. Hydr. Engrg., 130, pp. 580–585 (2004).Google Scholar
12.Tsai, T. L., Chiang, S. W. and Yang, J. C., “Characteristics Method with Cubic-Spline Interpolation for Open Channel Flow Computation,” Int. J. Numer. Methods in Fluids, 46, pp. 663–683 (2004).Google Scholar
13.Tsai, T. L. and Yang, J. C., “Kinematic Wave Modeling of Overland Flow Using Characteristics Method with Cubic-Spline Interpolation,” Advances in Water Resources, 28, pp. 661–670 (2005).Google Scholar
14.Tsai, T. L., Chiang, S. W. and Yang, J. C., “Examination on Characteristics Method with Cubic Interpolation for Advection-Diffusion Equation,” Computers and Fluids, 35, pp. 1217–1227 (2006).Google Scholar
15.Gerald, C. F. and Wheatley, P. O., Applied Numerical Analysis, Addison-Wesley, New York (2004).Google Scholar
16.Knott, G. D., Interpolating Cubic Splines, Birkhauser, Boston (1999).Google Scholar
17.Kvasov, B. I., Shape-Preserving Spline Application, Scientific Publishing, Singapore (2000).Google Scholar
18.DeBoor, C., A Practical Guide to Spline, Springer, New York (1978).Google Scholar
19.Morton, K. W. and Parrott, A. K., “Generalized Galerkin Methods for First-Order Hyperbolic Equation,” Journal of Computational Physics, 36, pp. 249–270 (1980).Google Scholar
20.Tsai, T. L., Yang, J. C. and Huang, L. H., “Hybrid Finite-Difference Scheme for Solving the Dispersion Equation,” Journal Hydraulic Engineering, 128, pp. 78–86 (2002).Google Scholar