A high order Path-Conservative Central-Upwind Arbitrary DERivative (PCCU-ADER) method for a generalized high order sediment transport model

An extension and numerical approximation of a sediment transport theory recently developed by Ngatcha and Nkonga (2023) [50] is considered. The model take into account the velocity fluctuation correlations to represent the effect of turbulence neglected in classical models based on shallow water equations. Then the model corrects the deficiency of the classical shallow water modeling in describing sediment transport phenomena. However, no numerical solution, mathematical and physical studies are available for this theory and the recent literature does not provide sufficient information about the turbulence modeling in shallow water context. A new second-order Path-Conservative Central-Upwind Arbitrary DERivative (PCCU-ADER for short) scheme is developed to approximate the model. The proposed second order scheme is proven to be stable, convergent, fast, well-balanced, preserving-positivity and shock-capturing. The benefits of our numerical scheme in comparison to those found in current literature (such as the Central-Upwind scheme, the HLL based Riemann solvers, etc.) are demonstrated through numerical and experimental validations. It has been demonstrated that turbulence emerges when water moves over abrupt topography and exerts an influence on sediment transport phenomena. Our findings indicate with this new hydrodynamic variable, the wavefront becomes too large and improve the classical wavefront description obtained by the shallow water equations.