How to make fully discrete entropy stability of three-point well-balanced finite-volume schemes

This paper presents a simple and robust method to enforce discrete entropy stability in first-order well-balanced finite volume schemes for systems of balance laws, including non-conservative terms. By leveraging an artificial viscosity technique originally introduced by Tadmor and extended in recent works, the authors propose an entropy-preserving modification applicable to a wide class of three-point schemes. The core contribution lies in the construction of a viscosity coefficient that maintains the well-balanced property while ensuring entropy dissipation at the discrete level. A theoretical framework is developed to guarantee robustness, well-balancing, and entropy stability under a suitable CFL condition. The approach is validated through numerical experiments, including applications to shallow water systems and two-layer flows, demonstrating accuracy and stability.