Modified Shallow Water Equations for significantly varying seabeds

Abstract

In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2016modifiedshallow,
  author = {Dutykh, D. and Clamond, D.},
  title = {Modified Shallow Water Equations for significantly varying seabeds},
  journal = {Appl. Math. Modelling},
  year = {2016},
  volume = {40},
  number = {23-24},
  pages = {9767--9787},
  doi = {10.1016/j.apm.2016.06.033},
  abstract = {In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.},
  keywords = {UNO scheme, Shallow water, Saint-Venant equations, finite volumes}
}