A fourth‐order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq‐type equations. Part I: model development and analysis

By R. Cienfuegos, E. Barthélemy, Philippe Bonneton. 2006

144 citations recorded by the source corpus.

A high‐order finite volume scheme is developed to numerically integrate a fully nonlinear and weakly dispersive set of Boussinesq‐type equations (the so‐called Serre equations) (J. Fluid Mech. 1987; 176:117–134; Surveys Geophys. 2004; 25(3–4):315–337). The choice of this discretization strategy is motivated by the fact that this particular set of equations is recasted in a convenient quasi‐conservative form. Cell face values are reconstructed using implicit compact schemes (J. Comput. Phys. 1999; 156:137–180; J. Comput. Phys. 2004; 198:535–566) and time integration is performed with the help of a four‐stage Runge–Kutta method. Numerical properties of the proposed scheme are investigated both, analytically using linear spectral analysis, and numerically for highly nonlinear cases. The numerical analysis indicates that the newly developed scheme has wider stability regions and better spectral resolution than most of the previously published numerical methods used to handle equivalent set of equations. Moreover, it was also noticed that the use of mixed‐order strategies to discretize convective and dispersive terms may result in an important overall reduction of the spectral resolution of the scheme. Additionally, there is some numerical evidence, which seems to indicate that the incorporation of a high‐order dispersion correction term as given by Madsen et al. (Coastal Eng. 1991; 15:371–388) may introduce instability in the system. Copyright © 2006 John Wiley & Sons, Ltd.

Subfields: Atmospheric Science, Computational Mechanics, Earth-Surface Processes

View Paper: https://www.semanticscholar.org/paper/57698b4880e92fcc6ca98b6399c66161d92161cb

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Citations in the corpus

A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations (2015, methodology): “While [4] proposed to do this using Strang splitting, we note that adding arbitrary source terms to the Saint-Venant system (or other systems of time-dependent partial differential equations) can often be done simply by including this source term in the appropriate stages of the time integration sch”
A new approach to handle wave breaking in fully non-linear Boussinesq models (2012, methodology): “…2 x uþ h∂ 2 x bu∂xu þ ∂xζ∂ 2 x bþ h 2 ∂3x b u2: ð4Þ α is an optimization parameter that should be taken equal to 1.159 in order to minimize the phase and group velocity errors in comparison with the linear Stokes theory (see the dispersion correctionmethod discussed in Cienfuegos et al. (2006)).”
Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation (2009, background): “[21] have shown that Serre equations can be formally established without assumption on the velocity profile.”