Advection‐Pressure Splitting Schemes Applied to a Non‐Conservative 1D Blood Flow Model With Transport for Arteries and Veins

We introduce new first order, splitting-based numerical schemes for the non-conservative one-dimensional (1D) blood flow equations with a general constant momentum correction coefficient that describe blood flow, for different velocity profiles, in arteries and veins with discontinuous mechanical and geometrical properties. In this model an advection equation for a passive scalar transport is also considered. Our schemes are inspired by the original flux vector splitting approach of Toro and Vázquez-Cendón (2012) designed for the Euler equations. They also represent an improvement of the work proposed by Toro et al. (2024) regarding non-conservative blood flow models, which considered a tube law describing only arteries, a momentum correction coefficient equal to one, no passive scalar transport and included a smaller number of discontinuous mechanical and geometrical parameters. The considered framework separates advection terms and pressure terms, generating two different systems of PDEs: the advection system in conservative form, and the pressure system in non-conservative form, both of which have a very simple eigenstructure compared to that of the full system. Our schemes involve approximate Riemann solvers and present a modification of the path-conservative framework that renders unnecessary the use of a path. They are systematically assessed on a carefully designed suite of test problems with exact solution and compared with several existing mainstream methods. A detailed efficiency analysis is performed in order to showcase the advantages of the proposed methodology in comparison with standard approaches.