The Lax-Friedrichs method in one-dimensional hemodynamics

TL;DR

This paper explores the application of the Lax-Friedrichs method to one-dimensional blood flow models, simplifying boundary condition implementation and enabling flexible coupling in vascular network simulations.

Contribution

It introduces a relaxation-based derivation of the scheme, extends it to second order, and demonstrates its effectiveness through numerical experiments.

Findings

The scheme is consistent and convergent.

Coupling conditions can be implemented without eigenstructure knowledge.

Second order MUSCL extension improves accuracy.

Abstract

The discretization of reduced one-dimensional hyperbolic models of blood flow using the Lax-Friedrichs method is discussed. Employing the well-established central scheme in this domain significantly simplifies the implementation of specific boundary and coupling conditions in vascular networks accounting e.g. for a periodic heart beat, vascular occlusions, stented vessel segments and bifurcations. In particular, the coupling of system extensions modeling patient specific geometries and therapies can be realized without information on the eigenstructure of the models. For the derivation of the scheme and the coupling conditions a relaxation of the model is considered and its discrete relaxation limit evaluated. Moreover, a second order MUSCL-type extensions of the scheme is introduced. Numerical experiments in uncoupled and coupled cases that verify the consistency and convergence of the approach are presented.