Flux vector splitting schemes applied to a conservative 1D blood flow model with transport for arteries and veins

Abstract

We present novel flux splitting-based numerical schemes for the 1D blood flow equations with an advection equation for a passive scalar, considering tube laws that allow to model blood flow in arteries and veins. Our schemes are inspired by the original flux vector splitting approach of Toro and Vázquez-Cendón (2012) and represent an extension of the work proposed by Toro et al. (2024), which addressed tube laws suitable for describing blood flow in arteries. Our schemes separate advection terms and pressure terms, generating two different systems of PDEs: the advection system and the pressure system, both of which have a very simple eigenstructure compared to that of the full system. We propose discretization schemes of the Godunov type that are simple and efficient. These qualities are evaluated on a suite of test problems with exact solution. A detailed efficiency analysis is performed in order to illustrate situations in which the proposed methodology results advantageous with respect to standard approaches.