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

## TL;DR

This paper develops high-order, well-balanced numerical schemes for one-dimensional blood flow models with discontinuous properties and source terms, effectively handling complex vessel networks and preserving steady states.

## Contribution

The paper introduces novel high-order semi-discrete methods that handle discontinuous parameters and algebraic source terms in blood flow models, extending to vessel networks.

## Key findings

- Methods outperform naive discretizations.
- Successfully handle discontinuities and source terms.
- Effective in complex vessel network simulations.

## Abstract

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. While discontinuities in model parameters are handled using the Generalized Hydrostatic Reconstruction, the presence of algebraic source terms implies that steady state solutions cannot always be computed analytically. In fact, steady states are defined by an ordinary differential equation that needs to be integrated numerically. Therefore, we resort on a numerical reconstruction operator to identify and, where appropriate, preserve steady states with an accuracy that depends on the reconstruction operator's numerical scheme. 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.