A Fully Well-balanced Scheme for the 1D Blood Flow Equations with Friction Source Term

We consider the 1D blood flow equations with friction source term. The main purpose of this work is the derivation of a positivity preserving and fully well-balanced scheme, which correctly approximates the solutions of this system, exactly preserves all the associated steady states, including the moving ones, and ensures the positivity of the cross-sectional area. To address such issues, a study of the moving steady states related to the friction source term is first performed. Afterwards, a Godunov-type scheme is constructed, by deriving a two-state approximate Riemann solver and by introducing a relevant discretization of the source term, to enforce the well-balanced property. The scheme is then adapted to the generalized model including a space-variable viscous resistance; a second-order well-balanced MUSCL extension of the scheme is also proposed. Numerical experiments are finally carried out in order to validate the scheme and highlight its properties.