A semi‐implicit finite element model for non‐hydrostatic (dispersive) surface waves

The objective of this research is to develop a model that will adequately simulate the dynamics of tsunami propagating across the continental shelf. In practical terms, a large spatial domain with high resolution is required so that source areas and runup areas are adequately resolved. Hence efficiency of the model is a major issue. The three-dimensional Reynolds averaged Navier–Stokes equations are depth-averaged to yield a set of equations that are similar to the shallow water equations but retain the non-hydrostatic pressure terms. This approach differs from the development of the Boussinesq equations where pressure is eliminated in favour of high-order velocity and geometry terms. The model gives good results for several test problems including an oscillating basin, propagation of a solitary wave, and a wave transformation over a bar. The hydrostatic and non-hydrostatic versions of the model are compared for a large-scale problem where a fault rupture generates a tsunami on the New Zealand continental shelf. The model efficiency is also very good and execution times are about a factor of 1.8 to 5 slower than the standard shallow water model, depending on problem size. Moreover, there are at least two methods to increase model accuracy when warranted: choosing a more optimal vertical interpolation function, and dividing the problem into layers. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  V. Casulli,et al.  Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow☆ , 1994 .

[2]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[3]  E. Barragy,et al.  Parallel iterative solution for h and p approximations of the shallow water equations , 1998 .

[4]  D. Peregrine.  Long waves on a beach , 1967, Journal of Fluid Mechanics.

[5]  G. Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves , 1995, Journal of Fluid Mechanics.

[6]  S. M. Holzer,et al.  Book Reviews : SYSTEM DYNAMICS Katsuhiko Ogata Prentice-Hall, Inc., Englewood Cliffs, NJ, 1978 , 1980 .

[7]  Roy A. Walters,et al.  An unstructured grid, three‐dimensional model based on the shallow water equations , 2000 .

[8]  Pengzhi Lin,et al.  A σ‐coordinate three‐dimensional numerical model for surface wave propagation , 2002 .

[9]  Roy A. Walters.  The frontal method in hydrodynamics simulations , 1980 .

[10]  Jurjen A. Battjes,et al.  Numerical simulation of nonlinear wave propagation over a bar , 1994 .

[11]  Marcel Zijlema,et al.  An accurate and efficient finite‐difference algorithm for non‐hydrostatic free‐surface flow with application to wave propagation , 2003 .

[12]  Von Schwind,et al.  Geophysical fluid dynamics for oceanographers , 1980 .

[13]  Vincenzo Casulli,et al.  A SEMI-IMPLICIT FINITE DIFFERENCE METHOD FOR NON-HYDROSTATIC, FREE-SURFACE FLOWS , 1999 .

[14]  P. Liu,et al.  A numerical study of submarine–landslide–generated waves and run–up , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  Numerical Simulation of Tsunami Generation, Propagation and Runup , 2004 .

[16]  Roy A. Walters,et al.  A robust, finite element model for hydrostatic surface water flows , 1998 .

[17]  E. Barragy,et al.  Application of Conjugate Gradient methods to tidal simulation , 1993 .

[18]  Vincenzo Casulli,et al.  Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems , 2002 .

[19]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[20]  Roy A. Walters,et al.  Geometrically based, automatic generator for irregular triangular networks , 1993 .