Derivation of a 2D PCCU-AENO method for nonconservative problems. Theory, Method and theoretical arguments

TL;DR

This paper develops a new 2D second-order path-conservative central-upwind scheme for complex sediment transport problems, demonstrating its accuracy, stability, and superiority through theoretical proofs and numerical tests.

Contribution

It introduces a novel 2D PCCU scheme with an AENO reconstruction for sediment transport modeling, including a new generalized shallow water equations framework.

Findings

The scheme is well-balanced and positivity-preserving.

It achieves second-order accuracy with the new AENO extension.

Numerical tests show superior performance over existing methods.

Abstract

In this paper, we introduce a methodology to design genuinely two-dimensional (2D) secondorder 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 resonance. This study is possible via a 2D 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 establish an existence theorem of global weak solutions. We show the convergence of a sequence of solutions of the proposed 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 by rigorous demonstrations that the derived 2D scheme on structured meshes is well-balanced and positivity-preserving. Several tests are made to show the ability and superb performance of the proposed numerical modeling. The results obtained are compared with those existing in the literature and with experimental data. The current modeling improves some recent results in sediment transport and shows a good ability to simulate sediment transport in large-range environments.