Upwind WENO scheme for Shallow Water Equations in contravariant formulation

An Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the Shallow Water Equations on generalized curvilinear coordinate systems is proposed. The Shallow Water Equations are expressed in a contravariant formulation in which Christoffel symbols are avoided. The equations are solved by using a high-resolution finite-volume method incorporated with an exact Riemann solver. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities on generalized boundary-conforming grids is presented; this procedure allows the numerical scheme to satisfy the freestream preservation property on highly-distorted grids. The proposed scheme ensures the satisfaction of the C-property. The model is verified against several benchmark tests, and the results are compared with theoretical and alternative numerical solutions.

► We propose an Upwind WENO scheme for contravariant Shallow Water Equations (SWEs). ► The presented contravariant formulation of SWE is free of Christoffel symbols. ► We present a procedure to correct numerical errors related to metric identities. ► The proposed scheme preserves freestream conditions on highly-distorted grids. ► The proposed scheme possesses good non-oscillatory and shock-capturing properties.