### Abstract

Free boundaries in shallow-water equations demarcate the time-dependent water line between ``flooded'' and ``dry'' topography. A novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous finite element discretization of the one-dimensional or symmetric shallow-water equations is presented. The algorithm uses fixed Eulerian flooded elements and one mixed Eulerian-Lagrangian element at each free boundary. The positivity of the mean water depth is ensured via a time step restriction based on analysis of a maximum principle for the discretized continuity equation while using an HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudo density (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. The approach is hypothesized to extend to finite-volume discretizations with similar mean level and slope reconstruction.

Original language | Undefined |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Publication status | Published - 2003 |

### Publication series

Name | |
---|---|

Publisher | Department of Applied Mathematics, University of Twente |

No. | 1683 |

ISSN (Print) | 0169-2690 |

### Keywords

- MSC-65M60
- IR-65868
- EWI-3503
- MSC-76M10

## Cite this

Bokhove, O. (2003).

*Flooding and drying in finite-element discretizations of shallow-water equations. Part 1: One dimension*. Enschede: University of Twente, Department of Applied Mathematics.