Volume 27, Issue 1
The Riemann Problem for a Blood Flow Model in Arteries

Wancheng Sheng, Qinglong Zhang & Yuxi Zheng

Commun. Comput. Phys., 27 (2020), pp. 227-250.

Published online: 2019-10

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  • Abstract

In this paper, the Riemann solutions of a reduced 6×6 blood flow model in medium-sized to large vessels are constructed. The model is non-strictly hyperbolic and non-conservative in nature, which brings two difficulties of the Riemann problem. One is the appearance of resonance while the other one is loss of uniqueness. The elementary waves include shock wave, rarefaction wave, contact discontinuity and stationary wave. The stationary wave is obtained by solving a steady equation. We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data. We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution. The Riemann solutions may contribute to the design of numerical schemes, which can apply to the complex blood flows.

  • Keywords

Blood flow, elementary waves, Riemann problem, non-uniqueness, global entropy condition.

  • AMS Subject Headings

35L65, 35L04, 58J45, 76N10, 35L60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mathwcsheng@t.shu.edu.cn (Wancheng Sheng)

zhangqinglong@shu.edu.cn (Qinglong Zhang)

zheng@psu.edu (Yuxi Zheng)

  • BibTex
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  • TXT
@Article{CiCP-27-227, author = {Sheng , Wancheng and Zhang , Qinglong and Zheng , Yuxi }, title = {The Riemann Problem for a Blood Flow Model in Arteries}, journal = {Communications in Computational Physics}, year = {2019}, volume = {27}, number = {1}, pages = {227--250}, abstract = {

In this paper, the Riemann solutions of a reduced 6×6 blood flow model in medium-sized to large vessels are constructed. The model is non-strictly hyperbolic and non-conservative in nature, which brings two difficulties of the Riemann problem. One is the appearance of resonance while the other one is loss of uniqueness. The elementary waves include shock wave, rarefaction wave, contact discontinuity and stationary wave. The stationary wave is obtained by solving a steady equation. We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data. We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution. The Riemann solutions may contribute to the design of numerical schemes, which can apply to the complex blood flows.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0220}, url = {http://global-sci.org/intro/article_detail/cicp/13320.html} }
TY - JOUR T1 - The Riemann Problem for a Blood Flow Model in Arteries AU - Sheng , Wancheng AU - Zhang , Qinglong AU - Zheng , Yuxi JO - Communications in Computational Physics VL - 1 SP - 227 EP - 250 PY - 2019 DA - 2019/10 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2018-0220 UR - https://global-sci.org/intro/article_detail/cicp/13320.html KW - Blood flow, elementary waves, Riemann problem, non-uniqueness, global entropy condition. AB -

In this paper, the Riemann solutions of a reduced 6×6 blood flow model in medium-sized to large vessels are constructed. The model is non-strictly hyperbolic and non-conservative in nature, which brings two difficulties of the Riemann problem. One is the appearance of resonance while the other one is loss of uniqueness. The elementary waves include shock wave, rarefaction wave, contact discontinuity and stationary wave. The stationary wave is obtained by solving a steady equation. We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data. We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution. The Riemann solutions may contribute to the design of numerical schemes, which can apply to the complex blood flows.

Wancheng Sheng, Qinglong Zhang & Yuxi Zheng. (2019). The Riemann Problem for a Blood Flow Model in Arteries. Communications in Computational Physics. 27 (1). 227-250. doi:10.4208/cicp.OA-2018-0220
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