Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme

A Finite Volume Well-balanced Weighted Essentially Non Oscillatory (WENO) scheme, fourth-order accurate in space and time, for the numerical integration of Shallow Water Equations with the bottom slope source term, is presented. The main novelty introduced in this work is a new method for managing bed discontinuities. This method is based on a suitable reconstruction of the conservative variables at the cell interfaces, coupled with a correction of the numerical flux based on the local conservation of total energy. Further changes regard the treatment of the source term, based on a high-order extension of the Divergence Form for Bed slope source term (DFB) method, and the application of an analytical inversion of the specific energy - depth relationship. Two ad hoc test cases, consisting of a steady flow over a step and a surge crossing a step, show the effectiveness of the method of treating bottom discontinuities. Several standard one-dimensional test cases are also used to verify the high-order accuracy, the C-property, and the good resolution properties of the resulting scheme, in the cases of both continuous and discontinuous bottoms. Finally, a comparison between the fourth-order scheme proposed here and a well-established second-order scheme emphasizes the improvement achieved using the higher-order approach.