CENTRAL WENO FOR SHALLOW WATER EQUATIONS IN CONTRAVARIANT FORMULATION

A new Central Weighted Essentially Non-Oscillatory scheme for the solution of the shallow water equations expressed directly in contravariant formulation is presented. The proposed central WENO scheme is the extension of the methodology presented by many authors(1,2,3,4) into the context defined by contravariant form of the equations. One of the most important elements of the C-WENO scheme based on this approach involves the advancing from time level t(n) to time level t(n+1) of the cell averaged values of flow variables. The extension of the above mentioned methodology into the contravariant environment implies that the contravariant shallow water equations must be expressed in integral form. An element of novelty presented in this paper regards the definition of a formal integral expression of the shallow water equations in contravariant formulation, in which Christoffel symbols are avoided. The WENO reconstructions are performed by a two dimensional interpolating procedure taking into account the curved coordinate lines; in the computational domain the spatial discretization step is constant: consequently the problems related to negative linear weights on unstructured meshes are overcome. The two dimensional reconstructions have a fifth-order spatial accuracy. A Natural Continuous Extension into a Runge-Kutta solver is involved in a fourth-order time discretization of motion equations. The proposed scheme ensures the satisfaction of the exact C-property.