Derivation of a 2D PCCU-AENO method for nonconservative problems

Abstract

In this paper, we introduce a methodology to design genuinely two-dimensional (2D) second-order well-balanced path-conservative central-upwind (PCCU) schemes. The scheme studies dam-break with high sediment concentration over abrupt moving topography quickly spatially variable even in the presence of the resonance. This study is possible via a 2D hyperbolic sediment transport model (including arbitrarily sloping sediment beds and associated energy and entropy) in new generalized Shallow Water equations derived with associated energy and entropy in this work. We expose some properties of the model and we establish an existence result of global weak solutions. In addition, we show the convergence of a sequence of solutions of the proposed new model. The second-order accuracy of the PCCU scheme is achieved using a new extension AENO (Averaging Essentially Non-Oscillatory) reconstruction developed in the 2D version of this work. We prove that the derived 2D scheme is well-balanced, positivity-preserving and steady states capturing. Several dam break tests are made to show the ability and the superb performances of the proposed numerical modeling. The proposed method can help to address many engineering problems.