On Well‐Balanced Finite Volume Methods for Nonconservative Nonhomogeneous Hyperbolic Systems

In this work we introduce a general family of finite volume methods for nonhomogeneous hyperbolic systems with nonconservative terms. We prove that all of them are “asymptotically well‐balanced”: they preserve all smooth stationary solutions in all the domain except for a set whose measure tends to zero as Δ x tends to zero. This theory is applied to solve the bilayer shallow‐water equations with arbitrary cross‐section. Finally, some numerical tests are presented for simplified but meaningful geometries, comparing the computed solution with approximated asymptotic analytical solutions.