A discontinuous Galerkin finite element model for river bed evolution under shallow flows

The accurate representation of morphodynamic processes and the ability to propagate changes in the riverbed over a wide range of space and time scales make the design and implementation of appropriate numerical schemes challenging. In particular, requirements of accuracy and stability for medium and long term simulations are difficult to meet. In this work, the derivation, design, and implementation of a discontinuous Galerkin finite element method (DGFEM) for sediment transport and bed evolution equations are presented. Numerical morphodynamic models involve a coupling between a hydrodynamic flow solver which acts as a driving force and a bed evolution model which accounts for sediment flux and bathymetry changes. A space DGFEM is presented based on an extended approach for systems of partial differential equations with non-conservative products, in combination with two intertwined Runge–Kutta time stepping schemes for the fast hydrodynamic and slow morphodynamic components. The resulting numerical scheme is verified by comparing simulations against (semi-)analytical solutions. These include the evolution of an initially symmetric, isolated bedform; the formation and propagation of a step in a straight channel due to a sudden overload of sediment discharge; the propagation of a travelling diffusive sediment wave in a straight channel; and, the evolution of an initially flat bed in a channel with a contraction. Finally, a comparison is made between a numerical simulation and field data of a trench excavated in the main channel of the Paraná river near Paraná city, Argentina.