Well balancing of the SWE schemes for moving-water steady flows

TL;DR

This paper introduces a new numerical scheme based on a modified HLLEM Riemann solver for the one-dimensional shallow water equations, which accurately reproduces moving-water steady flows and handles hydraulic jumps effectively.

Contribution

A novel scheme that exactly preserves total head and discharge in steady flows, improving accuracy over existing methods while discussing trade-offs in robustness and efficiency.

Findings

The new scheme accurately reproduces steady flows and dissipates energy correctly.

Comparison shows strengths in accuracy but reduced robustness and efficiency.

Proposed solutions improve robustness at the cost of increased complexity.

Abstract

In this work, the exact reproduction of a moving-water steady flow via the numerical solution of the one-dimensional shallow water equations is studied. A new scheme based on a modified version of the HLLEM approximate Riemann solver [Dumbser M. and Balsara D.S., J. Comput. Phys. 304 (2016) 275--319] that exactly preserves the total head and the discharge in the simulation of smooth steady flows and that correctly dissipates mechanical energy in the presence of hydraulic jumps is presented. This model is compared with a selected set of schemes from the literature, including models that exactly preserve quiescent flows and models that exactly preserve moving-water steady flows. The comparison highlights the strengths and weaknesses of the different approaches. In particular, the results show that the increase in accuracy in the steady-state reproduction is counterbalanced by a reduced robustness and numerical efficiency of the models. Some solutions to reduce these drawbacks, at the cost of increased algorithm complexity, are presented.