On well-balanced finite volume methods for non-conservative non-homogeneous hyperbolic systems

Abstract

In this work we introduce a general family of finite volume methods for non-homogeneous hyperbolic systems with non-conservative terms. We prove that all of them are “asymptotically well-balanced”: They preserve all smooth stationary solutions in all the domain but a set whose measure tends to zero as ∆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.