Central WENO scheme for the integral form of contravariant shallow‐water equations

Abstract

AbstractA new Central Weighted Essentially Non‐Oscillatory scheme for the solution of the shallow‐water equations expressed in contravariant formulation is presented. One of the most efficient methodologies belonging to Central WENO family involves: reconstructions of cell‐averaged values of flow variables, reconstruction of point‐values of flow variables, advancing from time level tn to time level tn+1 of the cell‐averaged values. The extension of the above‐mentioned methodology into the contravariant environment implies that the contravariant shallow‐water equations must be expressed in an integral form. An element of novelty presented in this paper regards the definition of a formal integral expression of the shallow‐water equations in a contravariant formulation, in which the Christoffel symbols are avoided. The WENO reconstructions are performed by a two‐dimensional interpolating procedure taking into account the curved coordinate lines. The proposed scheme ensures the satisfaction of the exact C‐property. Several two‐dimensional test cases are used to verify the good resolution for smooth and discontinuous solutions. Copyright © 2010 John Wiley & Sons, Ltd.