On devising Boussinesq-type models with bounded eigenspectra: One horizontal dimension
Journal article, 2014

The propagation of water waves in the nearshore region can be described by depthintegrated Boussinesq-type equations. The dispersive and nonlinear characteristics of the equations are governed by tuneable parameters. We examine the associated linear eigenproblem both analytically and numerically using a spectral element method of arbitrary spatial order p. It is shown that existing sets of parameters, found by optimising the linear dispersion relation, give rise to unbounded eigenspectra which govern stability. For explicit time-stepping schemes the global CFL time-step restriction typically requires Delta t proportional to p(-2). We derive and present conditions on the parameters under which implicitlyimplicit Boussinesq-type equations will exhibit bounded eigenspectra. Two new bounded versions having comparable nonlinear and dispersive properties as the equations of Nwogu (1993) and Schaffer and Madsen (1995) are introduced. Using spectral element simulations of stream function waves it is illustrated that (i) the bounded equations capture the physics of the wave motion as well as the standard unbounded equations, and (ii) the bounded equations are computationally more efficient when explicit time-stepping schemes are used. Thus the bounded equations were found to lead to more robust and efficient numerical schemes without compromising the accuracy.

Time integration

Spectral/hp element method

Implicitly-implicit equations

Eigenvalue analysis

Nonlinear dispersive water waves

Boussinesq-type equations

Author

Claes Eskilsson

Chalmers, Shipping and Marine Technology, Division of Marine Design

A. P. Engsig-Karup

Technical University of Denmark (DTU)

Journal of Computational Physics

0021-9991 (ISSN) 1090-2716 (eISSN)

Vol. 271 261-280

Subject Categories

Marine Engineering

DOI

10.1016/j.jcp.2013.08.048

More information

Latest update

2/28/2018