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A well‐balanced positivity‐preserving central‐upwind scheme for one‐dimensional blood flow models
International Journal for Numerical Methods in Fluids ( IF 1.8 ) Pub Date : 2020-06-29 , DOI: 10.1002/fld.4887
Gerardo Hernandez‐Duenas 1 , Guillermo Ramirez‐Santiago 1
Affiliation  

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.

中文翻译:

一维血流模型的平衡良好的保持阳性的中央逆风方案

在这项工作中,我们考虑了通过顺应性轴对称倾斜血管的血流的双曲一维(1D)模型。压力是横截面积和其他模型参数的函数。该模型的重要特征通过数值方案在离散级别上继承。例如,稳态的存在可以以低计算成本提供关于流动特性的重要信息。在这里,我们通过保持不变的数量来表征一大类平滑均衡解。在离散级别,当稳态受到扰动时,数值方案中的均衡特性会导致结果准确。另一方面,模型配备了熵函数和熵不等式,可以帮助我们选择与物理相关的弱解。描述了一大类半离散熵满足数值方案。另外,实现了横截面积的阳性保持。数值结果表明,该方案具有鲁棒性,稳定性和准确性。本文的最终目标是从医学角度将数值应用到更相关的案例中。特别地,具有适当参数的心动周期的数值模拟表明,增加动脉壁的刚度会延迟冲击波的形成。还测量了倾斜血管中的重力效应,并使用理想主动脉模型进行了模拟。本文的最终目标是从医学角度将数值应用到更相关的案例中。特别地,具有适当参数的心动周期的数值模拟表明,增加动脉壁的刚度会延迟冲击波的形成。还测量了倾斜血管中的重力效应,并使用理想主动脉模型进行了模拟。本文的最终目标是从医学角度将数值应用到更相关的案例中。特别地,具有适当参数的心动周期的数值模拟表明,增加动脉壁的刚度会延迟冲击波的形成。还测量了倾斜血管中的重力效应,并使用理想主动脉模型进行了模拟。
更新日期:2020-06-29
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