Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant-Exner morphodynamic model

TL;DR

This paper introduces an analytical eigenstructure formulation for the DOT Riemann solver applied to the de Saint Venant-Exner model, significantly improving computational efficiency and stability in morphodynamic simulations.

Contribution

The work develops a new analytical eigenstructure approach for the DOT scheme, reducing computational cost and enhancing stability for the de Saint Venant-Exner model.

Findings

A-DOT is more efficient than original DOT and PRICE-C methods at the same error level.

Analytical eigenstructure improves numerical stability and reduces computation time.

Method performs well in practical case studies with experimental data.

Abstract

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant-Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with an analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.