Stability of a Cartesian grid projection method for zero Froude number shallow water flows

  • Authors:

  • Stefan Vater;Rupert Klein

  • Affiliations:

  • Freie Universität (FU) Berlin, Numerische Mathematik/Scientific Computing, FB Mathematik und Informatik, Institut  für Mathematik, Arnimallee 6, 14195, Berlin, Germany;Freie Universität (FU) Berlin, Numerische Mathematik/Scientific Computing, FB Mathematik und Informatik, Institut  für Mathematik, Arnimallee 6, 14195, Berlin, Germany

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2009

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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.