The entropy fix in augmented Riemann solvers in presence of source terms: Application to the Shallow Water Equations

Extensions to the Roe and HLL method have been previously formulated in order to solve the Shallow Water equations in the presence of source terms. These were named the Augmented Roe (ARoe) method and the HLLS method, respectively. This paper continues developing these formulations by examining how entropy corrections can be appropriately fitted in for the ARoe method and how the HLLS method can be formulated more generally. This is done in two ways. Firstly, this paper extends the reasoning of Harten and Hyman required by the ARoe method to include the source term contributions and thus arrives to a more complete formulation of the entropy fix, which will be compared with the approximation presented in previous works through numerical experiments. Secondly, it is shown how a relaxation of the criteria used when choosing waves in the HLLS method yields better solutions to problems where the HLLS would previously fail. In summary, this paper seeks to offer a comprehensible review of the ARoe and HLLS methods while improving its performance in cases with transcritical rarefactions for the inhomogeneous Shallow Water Equations in one dimension.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This paper extends the ARoe and HLLS methods for solving the Shallow Water equations with source terms. It proposes more complete entropy correction formulas and a more general formulation of the HLLS method. Experimental results show that these methods are more effective in solving specific problems.