Comparison between different methods to compute the numerical fluctuations in path-conservative schemes for SWE-Exner model

We consider the Shallow Water Equations (SWE) coupled with the Exner equation. To solve these balance laws, we implement a P0P2-ADER scheme using a path conservative method for handling the non-conservative terms of the system. In this framework we present a comparison between three different Dumbser-Osher-Toro (DOT) Riemann solvers. In particular, we focus on three different approaches to obtain the eigensystem of the Jacobian matrix needed to compute the fluctuations at the cell edges. For a general formulation of the bedload transport flux, we compute eigenvalues and eigenvectors numerically, analytically and using an approximate original solution for lowland rivers (i.e. with Froude number Fr << 1) based on a perturbative analysis. To test these different approaches we use a suitable set of test cases. Three of them are presented here: a test with a smooth analytical solution, a Riemann problem with analytical solution and a test in which the Froude number approaches unity. Finally, a computational costs analysis shows that, even if the approximate DOT is the most computationally efficient, the analytical DOT is more robust with about 10% of additional cost. The numerical DOT is shown to be the heavier solution.