Analytical Evaluation of Amplification Factors, Stability, and Error Analysis of Square Finite Element Solution for the Kinematic Wave Shallow Water Equations
Publication: Journal of Hydrologic Engineering
Volume 19, Issue 9
Abstract
A generalized error analysis is discussed in this paper to investigate solution accuracy for watershed simulation results using overland flow equations. The total or compounded error for watershed simulations may increase with increasing spatial resolution, scale, and number of iterations to convergence if inaccurate time-step is chosen for the numerical simulations. A relationship is derived between watershed scale error, overland flow plane error, finite element, and node errors. The scale versus compounded error relationship for watershed scale simulations is a function of element spatial discretization level, dynamic time-step criteria, and the relationship between element and nodal errors. The Courant number and wave number space solution is simulation optimal and would require separate solutions of Fourier-transform (and its inversion) of finite element nodal equations to obtain solution accuracy within 5%. Dynamic FEM square grid discretization criteria is derived from the results of eigenvalue analysis and dynamic time-step results for various finite element schemes—consistent, lumped, and upwind. Courant number and amplification factor evolutions have been determined for a reference case for consistent, lumped, and upwind finite element schemes. The number of amplification factors that are greater than 1.0 essentially determine the reason why oscillations could be noticed in numerical simulations even though the Courant number is less than 1.0 for few of the finite element numerical schemes.
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Acknowledgments
This work was supported by a USGS grant through the Indiana Water Resources Center and Department of Science and Technology (DST, Government of India) grant (SR/S3/MERC-0001/2011) of the project titled “Fourier Analysis and Development of Two-Dimensional Finite Element Schemes for Shallow Water Equations and Transport related Systems.” The authors would also like to thank the editor and reviewers for positive criticism and for overall suggestions made in improving the quality of the paper.
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Published In
Journal of Hydrologic Engineering
Volume 19 • Issue 9 • September 2014
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Jan 9, 2013
Accepted: Dec 2, 2013
Published online: Dec 4, 2013
Published in print: Sep 1, 2014
Discussion open until: Nov 17, 2014
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