On the well-balance property of Roe?s method for nonconservative hyperbolic systems

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Godunov method for nonconservative hyperbolic systems (2007, background): “According to [21], given a curve γ ∈ Γ, a numerical scheme is said to be exactly well-balanced (respectively well-balanced with order k) for γ if it solves exactly (respectively up to the order O(∆x)) regular stationary solutions W satisfying W (x) ∈ γ for every x.”
A CENTRAL SCHEME FOR SHALLOW WATER FLOWS ALONG CHANNELS WITH IRREGULAR GEOMETRY (2009, unknown): “Upwind schemes for the treatment of balance laws include [4,6,17,21], and kinetic approaches include [2,22].”
Numerical methods for nonconservative hyperbolic systems: a theoretical framework (2006, methodology): “In [29] it has been shown that a Roe scheme based on a family of paths Ψ is exactly well-balanced for a curve γ ∈ Γ if, given two states WL and WR in γ, the path Ψ(s;WL,WR) is a parameterization of the arc of γ linking these states.”