A splitting scheme for the coupled Saint-Venant-Exner model

TL;DR

This paper introduces a simple, accurate, and robust splitting numerical scheme for the Saint-Venant-Exner model, enabling efficient simulation of bed evolution in shallow water systems with flexible sediment transport modeling.

Contribution

It develops a novel splitting method combining Godunov-type schemes for coupled conservative and non-conservative subsystems, extending to second order accuracy with ADER and AENO techniques.

Findings

The scheme is highly accurate and robust in various test cases.

It maintains well-balanced properties and flexibility for sediment transport formulas.

The method simplifies implementation compared to traditional Godunov approaches.

Abstract

We present a splitting method for the one-dimensional Saint-Venant-Exner equations used for describing the bed evolution in shallow water systems. We adapt the flux vector splitting approach of Toro and Vazquez-Cend\`on and identify one subsystem of conservative equations (advection system) and one of non-conservative equations (pressure system), both having a very simple eigenstructure compared to the full system. The final numerical scheme is constructed using a Godunov-type path-conservative scheme for the pressure system and a simple conservative Godunov method for the advection system and solved following a coupled solution strategy. The resulting first-order accurate method is extended to second order of accuracy in space and time via the ADER approach together with an AENO reconstruction technique. Accuracy, robustness and well-balanced properties of the resulting scheme are assessed through a carefully selected suite of testcases. The scheme is exceedingly simple, accurate and robust as the sophisticated Godunov methods. A distinctive feature of the novel scheme is its flexibility in the choice of the sediment transport closure formula, which makes it particularly attractive for scientific and engineering applications.