Accuracy Order of Crank–Nicolson Discretization for Hydrostatic Free-Surface Flow
Publication: Journal of Engineering Mechanics
Volume 130, Issue 8
Abstract
Application of Crank–Nicolson (CN) discretization to the hydrostatic (or shallow-water) free-surface equation in two-dimensional or three-dimensional Reynolds-averaged Navier–Stokes models neglects a second order term. The neglected term is zero at steady state, so it does not appear in steady-state accuracy analyses. A new correction term is derived that restores second-order accuracy. The correction is significant when the amplitude of the surface oscillation is within two orders of magnitude of the water depth and the barotropic Courant–Friedrichs–Lewy (CFL) stability condition is less than unity. Analysis shows that the CN accuracy for an unforced free-surface oscillation is degraded to first order when the barotropic CFL stability condition is greater than unity, independent of whether or not the new correction term is applied. The results indicate that the semi-implicit Crank–Nicolson method, applied to the hydrostatic free-surface evolution equation, is only first-order accurate for the time and space scales typically used in lake, estuarine, and coastal ocean studies.
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Information
Published In
Journal of Engineering Mechanics
Volume 130 • Issue 8 • August 2004
Pages: 904 - 910
Copyright
Copyright © 2004 American Society of Civil Engineers.
History
Received: Aug 11, 2003
Accepted: Dec 23, 2003
Published online: Jul 15, 2004
Published in print: Aug 2004
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