On the inaccuracies of some finite volume discrtizations of the linearized shallow water problem
S. Faure 1, M. Petcu 2, R. Temam 3, J. Tribbia 41 Laboratoire de Mathematiques, Universite Paris-Sud, France.
2 Laboratoire de Mathematiques et Applications, UMR 6086, Universite de Poitiers, France; and The Institute of Mathematics of the Romanian Academy, Bucharest, Romania
3 The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, USA.
4 National Center for Atmospheric Research, Boulder, USA.
Received by the editors February 10, 2011.
In this article we are interested in the study of the errors introduced by different finite volume discretizations in the study of the wave frequencies. This study is made in the context of hyperbolic systems arising in meteorology and oceanography. We show the existence of significant errors in the dispersion relation, only the long waves being computed accurately; this conclusion is similar to the finite differences case described in the article of Grotjahn and O¡¦Brien quoted below. For the case of inertia-gravity waves, we study three often used schemes which are based on the upwind, centered or Lax-Wendroff fluxes. Moreover, the Total Variation Diminishing method (TVD) made from these fluxes (which usually provides an efficient way to eliminate spurious numerical oscillations) will give the same errors in the dispersion relation.AMS subject classifications: 35L60, 35Q35, 76B15, 76B65
Key words: finite volume discretizations, spurious caustics, numerical error, dispersion relation, hyperbolic systems.