High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties, friction and gravity

We present well-balanced, high-order, semi-discrete numerical schemes for one-dimensional blood flow models with discontinuous mechanical properties and algebraic source terms representing friction and gravity. The design of these schemes relies on the computation of stationary solutions, which are incorporated into a reconstruction procedure. However, this poses two main difficulties. On the one hand, discontinuities in the model parameters lead to nonconservative products in the source terms, giving rise to both theoretical and numerical challenges. On the other hand, the presence of friction and gravity terms prevents obtaining analytical expressions of the stationary solutions, neither explicitly nor implicitly. To overcome these issues, we employ the Generalized Hydrostatic Reconstruction to handle discontinuities, and a collocation method to numerically approximate the stationary solutions required in the reconstruction. To the best of our knowledge, this is the first time that both techniques are combined in the context of systems of balance laws. We extend our methods to deal with networks of vessels and show numerical results for single- and multiple-vessel tests, including a network of 118 vessels, demonstrating the capacity of the presented methods to outperform naive discretizations of the equations under study.