Vater, S. and Klein, R.
(2009)
*Stability of a Cartesian grid projection method for zero Froude number shallow water flows.*
Numerische Mathematik, 113
(1).
pp. 123-161.

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## Abstract

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test functions. The key innovation compared to existing finite volume projection methods is a correction of the in-cell slopes of the momentum by the second projection. The stability of the projection is proved using a generalized theory for mixed finite elements. In order to do so, the validity of three different inf-sup conditions has to be shown. The results of preliminary numerical test cases demonstrate the method's applicability. On fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme.

Item Type: | Article |
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Uncontrolled Keywords: | Finite Volumes, Projection Methods, Shallow Water Equations |

Subjects: | Mathematical and Computer Sciences > Mathematics > Numerical Analysis |

Divisions: | Department of Mathematics and Computer Science > Institute of Mathematics > Geophysical Fluid Dynamics Group |

ID Code: | 537 |

Deposited By: | Ulrike Eickers |

Deposited On: | 13 Jul 2009 13:07 |

Last Modified: | 15 Jul 2009 14:48 |

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