A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations

In this paper, we introduce a novel approach for constructing robust, fully well-balanced numerical methods for the one-dimensional Saint-Venant system, both with and without the Manning friction term. Inspired by the method presented in [R. Abgrall, Commun. Appl. Math. Comput., 5 (2023), pp. 370–402], we first combine the conservative and primitive formulations of the studied hyperbolic system in a natural way. The solution is globally continuous and described by a combination of point and average values. The point and average values will then be evolved by two different forms of the studied Saint-Venant system, and we will show how to deal with the yield conservative and primitive forms in a well-balanced manner. The developed schemes are capable of exactly preserving both the still-water and the moving-water equilibria. We demonstrate the behavior of the proposed new scheme on several challenging examples. © 2024 Society for Industrial and Applied Mathematics.