Time-Centered Split Method for Implicit Discretization of Unsteady Advection Problems
Publication: Journal of Engineering Mechanics
Volume 135, Issue 4
Abstract
A new method for implicit solution of unsteady nonlinear advection equations is presented. This time-centered split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically nonlinear coupled equations for any number of variables. The TCS algorithm is compared to other nonlinear solution methods (local linearization, Picard iteration, and Newton iteration) and applied to the Crank-Nicolson discretization of the one-dimensional Burgers’ equation. The temporal accuracy and practical stability of the method is confirmed using an unsteady flow test case with an analytical solution. The method is shown to require computational effort similar to local linearization, but does not require discrete computation of a functional Jacobian for solution.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writers would like to thank Dr. Spyros Kinnas and Dr. Graham Carey as well as Yi-Hsiang Yu, Vimal Vinayan, and Paula Kulis for insightful discussions and suggestions. The writers also appreciate all the valuable comments from the anonymous reviewers. This research was partially supported by NSF Earth Science Hydrology Grant No. NSFEAR-0710901 and Texas Water Development Board Research and Planning Grant Nos. UNSPECIFIED2003-469 and UNSPECIFIED2004-483. Additional support was provided by the Center for Research in Water Resources at the University of Texas at Austin.
References
Blumberg, A. F., and Mellor, G. L. (1987). “A description of a three-dimensional coastal ocean circulation model.” Three-dimensional coastal ocean models, American Geophysical Union, Washington, D.C.
Bourchtein, A., and Bourchtein, L. (2007). “Semi-Lagrangian semi-implicit time-splitting scheme for the shallow water equations.” Int. J. Numer. Methods Fluids, 54(4), 453–471.
Casulli, V. (1990). “Semi-implicit finite difference methods for the two-dimensional shallow water equations.” J. Comput. Phys., 86(1), 56–74.
Dubois, T., Jauberteau, F., Temam, R. M., and Tribbia, J. (2005). “Multilevel schemes for the shallow water equations.” J. Comput. Phys., 207(2), 660–694.
Ferziger, J. H., and Peric, M. (1999). Computational methods for fluid dynamics, Springer, New York.
Fujima, K., and Shigemura, T. (2000). “Determination of grid size for leap-frog finite difference model to simulate tsunamis around a conical island.” Coastal Eng., 42(2), 197–210.
Kadalbajoo, M. K., and Awasthi, A. (2006). “A numerical method based on Crank-Nicolson scheme for Burgers’ equation.” Appl. Math. Comput., 182(2), 1430–1442.
Kar, S. K. (2006). “A semi-implicit Runge-Kutta time-difference scheme for the two-dimensional shallow water equations.” Mon. Weather Rev., 134, 2916–2926.
Kutluay, S., Bahadir, A. R., and Ozdes, A. (1999). “Numerical solution of one-dimensional Burgers equation: Explicit and exact-explicit finite difference methods.” J. Comput. Appl. Math., 103(2), 251–261.
Lomax, H., Pulliam, T. H., and Zingg, D. W. (1999). Fundamentals of computational fluid dynamics, Springer, New York.
Paniconi, C., and Putti, M. (1994). “A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems.” Water Resour. Res., 30(12), 3357–3374.
Pozrikidis, C. (2005). Introduction to finite and spectral element methods using MATLAB, Chapman and Hall/CRC, Boca Raton, Fla.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1992). Numerical recipes in FORTRAN: The art of scientific computing, Cambridge University Press, New York.
Rueda, F. J., and Schladow, G. S. (2002). “Quantitative comparison of models for barotropic response of homogeneous basins.” J. Hydraul. Eng., 128(2), 201–213.
Tannehill, J. C., Anderson, D. A., and Pletcher, R. H. (1997). Computational fluid mechanics and heat transfer, Taylor and Francis, Washington, D.C.
Information & Authors
Information
Published In
Copyright
© 2009 ASCE.
History
Received: May 8, 2007
Accepted: Nov 25, 2008
Published online: Apr 1, 2009
Published in print: Apr 2009
Notes
Note. Associate Editor: Brett F. Sanders
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.