Stability and consistency of nonhydrostatic free-surface models using the semi-implicit θ-method


Correspondence to: Sean Vitousek, Department of Civil and Environmental Engineering, The Jerry Yang and Akiko Yamazaki Environment and Energy Building, 473 Via Ortega, Office M-14, Stanford, CA 94305, USA.



The θ-method is a popular semi-implicit finite-difference method for simulating free-surface flows. Problem stiffness, arising because of the presence of both fast and slow timescale processes, is easily handled by the θ-method. In most ocean, coastal, and estuary modeling applications, stiffness is caused by fast surface gravity wave timescales imposed on slower timescales of baroclinic variability. The method is well known to be unconditionally stable for shallow water (hydrostatic) models when math formula, where θ is the implicitness parameter. In this paper, we demonstrate that the method is also unconditionally stable for nonhydrostatic models, when math formula for both pressure projection and pressure correction methods. However, the methods result in artificial damping of the barotropic mode. In addition to investigating stability, we also estimate the form of artificial damping induced by both the free surface and nonhydrostatic pressure solution methods. Finally, this analysis may be used to estimate the damping or growth associated with a particular wavenumber and the overall order of accuracy of the discretization. Copyright © 2012 John Wiley & Sons, Ltd.