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A well‐balanced positivity‐preserving central‐upwind scheme for one‐dimensional blood flow models
Author(s) -
HernandezDuenas Gerardo,
RamirezSantiago Guillermo
Publication year - 2021
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4887
Subject(s) - upwind scheme , mathematics , computer simulation , entropy (arrow of time) , numerical analysis , finite volume method , mathematical optimization , mathematical analysis , mechanics , physics , statistics , quantum mechanics , discretization
Summary In this work, we consider a hyperbolic one‐dimensional (1D) model for blood flow through compliant axisymmetric tilted vessels. The pressure is a function of the cross‐sectional area and other model parameters. Important features of the model are inherited at the discrete level by the numerical scheme. For instance, the existence of steady states may provide important information about the flow properties at low computational cost. Here, we characterize a large class of smooth equilibrium solutions by means of quantities that remain invariant. At the discrete level, the well‐balanced property in the numerical scheme leads to accurate results when steady states are perturbed. On the other hand, the model is equipped with an entropy function and an entropy inequality that can help us choose the physically relevant weak solutions. A large class of semidiscrete entropy‐satisfying numerical schemes is described. In addition, preservation of positivity for the cross‐sectional area is achieved. Numerical results show the scheme is robust, stable, and accurate. The ultimate goal of this article is the numerical application to cases that are more relevant from the medical viewpoint. In particular, a numerical simulation of cardiac cycles with appropriate parameters shows that increasing the rigidity of the artery walls delays the formation of shock waves. Gravity effects are also measured in tilted vessels, and a simulation using an idealized aorta model was conducted.