New Well-Balanced Path-Conservative Numerical Scheme for a Partially Relaxed Two-Layer Hydro-Sediment-Morphodynamic Model

Abstract

By incorporating horizontal sediment transport and bed erosion into the two-layer depth-averaged shallow water system, one can construct a hydro-sediment-morphodynamic mathematical model in which a variable sediment concentration in a denser layer and interaction between underflows and ambient water are taken into account. Numerically solving such a promising system presents challenges due to its inherent mathematical properties of being conditionally hyperbolic and nonconservative. These properties may result in instability and incorrect results near sharp hydraulic gradients in numerical solutions. The major contribution of this paper is to develop a novel numerical scheme to overcome these issues and to allow the system to be extended to a wider range of engineering applications. To this end, for a one-dimensional two-layer hydro-sediment-morphodynamic system, (1) its mathematical model is first reformulated to a novel relaxation format, which is mathematically equivalent and unconditionally hyperbolic, by relaxing interlayer surface level, and (2) this relaxation system is then solved using a novel path-conservative numerical algorithm with new discrete formulas for the fluxes and nonconservative terms. Thus, the proposed numerical scheme ensures the well-balanced property, and has a major advantage in its ability to accurately and stably handle sharp hydraulic/density gradients with second-order accuracy. Several numerical and laboratory tests are conducted to demonstrate the performance of the proposed new scheme.