Well-balanced discontinuous Galerkin methods for the one-dimensional blood flow through arteries model with man-at-eternal-rest and living-man equilibria. (English) Zbl 1519.76139
Summary: The movement of blood flow in arteries can be modeled by a system of conservation laws and has a range of applications in medical contexts. In this paper, we present efficient well-balanced discontinuous Galerkin methods for the one-dimensional blood flow model, which preserve the man-at-eternal-rest (zero velocity) and more general living-man (non-zero velocity) equilibria. Recovery of well-balanced states, decomposition of the numerical solutions into the equilibrium and fluctuation parts, and appropriate source term and numerical flux approximations are the key ideas in designing well-balanced methods. Numerical examples are presented to verify the well-balanced property, high order accuracy, good resolution for both smooth and discontinuous solutions, and the ability to capture nearly equilibrium solutions well. We also test the proposed methods on nearly equilibrium flows with various Shapiro numbers. Man-at-eternal-rest well-balanced methods work well for problems with low Shapiro number, but generate spurious flows when Shapiro number gets larger, while the living-man well-balanced methods perform well for all ranges of Shapiro number.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76Z05 | Physiological flows |
| 92C35 | Physiological flow |
Keywords:
well-balanced methods; discontinuous Galerkin methods; blood flow through arteries; steady state solutions; Shapiro number; nearly equilibrium flowReferences:
| [1] | Audusse, E.; Bouchut, F.; Bristeau, M.-O.; Klein, R.; Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J Sci Comput, 25, 2050-2065 (2004) · Zbl 1133.65308 |
| [2] | Bermudez, A.; Vazquez, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comput Fluids, 23, 1049-1071 (1994) · Zbl 0816.76052 |
| [3] | Boileau, E.; Nithiarasu, P.; Blanco, P.; Müller, L.; Fossan, F.; Hellevik, L., A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling, Int J Numer Method Biomed Eng, 31, 10, e02732 (2015) |
| [4] | Buli, J.; Xing, Y., A discontinuous Galerkin method for the Aw-Rascle traffic flow model on networks, J Comput Phys, 406, 109183 (2020) · Zbl 1453.65310 |
| [5] | Čanić, S., Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties, Comput Vis Sci, 4, 3, 147-155 (2002) · Zbl 0987.92011 |
| [6] | Čanić, S.; Piccoli, B.; Qiu, J.-M.; Ren, T., Runge-Kutta discontinuous galerkin method for traffic flow model on networks, J Sci Comput, 63, 1, 233-255 (2015) · Zbl 1321.90034 |
| [7] | Chandrashekar, P.; Klingenberg, C., A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J Scientif Comput, 37, 382-402 (2015) · Zbl 1320.76078 |
| [8] | Delestre, O.; Lagrée, P.-Y., A well-balanced finite volume scheme for blood flow simulation, Int J Numer Methods Fluids, 72, 2, 177-205 (2013) · Zbl 1455.76215 |
| [9] | Euler, L., Principia pro motu sanguinis per arterias determinando, Opera Postuma, 2, 814-823 (1862) |
| [10] | Formaggia, L.; Gerbeau, J.-F.; Nobile, F.; Quarteroni, A., Numerical treatment of defective boundary conditions for the navier-Stokes equations, SIAM J Numer Anal, 40, 1, 376-401 (2002) · Zbl 1020.35070 |
| [11] | Formaggia, L.; Gerbeau, J.; Nobile, F.; Quarteroni, A., On the coupling of 3D and 1D navier-Stokes equations for flow problems in compliant vessels, Comput Methods Appl Mech Eng, 191, 6, 561-582 (2001) · Zbl 1007.74035 |
| [12] | ISBN 978-3-642-56288-4. · Zbl 1001.76127 |
| [13] | Formaggia, L.; Quarteroni, A.; Veneziani, A., Cardiovascular mathematics: modeling and simulation of the circulatory system (2009), Springer-Verlag Italia, Milano · Zbl 1300.92005 |
| [14] | Ghigo, A.; Delestre, O.; Fullana, J.-M.; Lagrée, P.-Y., Low-shapiro hydrostatic reconstruction technique for blood flow simulation in large arteries with varying geometrical and mechanical properties, J Comput Phys, 331, 108-136 (2017) · Zbl 1378.76151 |
| [15] | Kurganov, A., Finite-volume schemes for shallow-water equations, Acta Numerica, 27, 289-351 (2018) · Zbl 1430.76372 |
| [16] | Li, G.; Delestre, O.; Yuan, L., Well-balanced discontinuous galerkin method and finite volume weno scheme based on hydrostatic reconstruction for blood flow model in arteries, Int J Numer Methods Fluids, 86, 7, 491-508 (2018) |
| [17] | Li, G.; Xing, Y., Well-balanced discontinuous galerkin methods with hydrostatic reconstruction for the euler equations with gravitation, J Comput Phys, 352, 445-462 (2018) · Zbl 1375.76089 |
| [18] | ISBN 0521216893 0521292336, http://www.loc.gov/catdir/toc/cam027/77008174.html. · Zbl 0153.30201 |
| [19] | Murillo, J.; García-Navarro, P., A roe type energy balanced solver for 1D arterial blood flow and transport, Comput Fluids, 117, 149-167 (2015) · Zbl 1390.76941 |
| [20] | http://www.sciencedirect.com/science/article/pii/S0021999113001277. · Zbl 1323.92066 |
| [21] | Müller, L.; Toro, E. F., Well-balanced high-order solver for blood flow in networks of vessels with variable properties, Int J Numer Method Biomed Eng, 29, 12, 1388-1411 (2013) |
| [22] | Noelle, S.; Xing, Y.; Shu, C.-W., High order well-balanced finite volume WENO schemes for shallow water equation with moving water, J Comput Phys, 226, 29-58 (2007) · Zbl 1120.76046 |
| [23] | Olufsen, M. S.; Peskin, C. S.; Kim, W. Y.; Pedersen, E. M.; Nadim, A.; Larsen, J., Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Ann Biomed Eng, 28, 11, 1281-1299 (2000) |
| [24] | Pedley, T. J., The fluid mechanics of large blood vessels (1980), Cambridge University Press · Zbl 0449.76100 |
| [25] | Pontrelli, G., A mathematical model of flow in a liquid-filled visco-elastic tube, Med Biol Eng Comput, 40, 550-556 (2002) |
| [26] | ISBN 978-3-662-04784-2. |
| [27] | Quarteroni, A.; Tuveri, M.; Veneziani, A., Computational vascular fluid dynamics: problems, models and methods, Comput Vis Sci, 2, 4, 163-197 (2000) · Zbl 1096.76042 |
| [28] | Sherwin, S. J.; Formaggia, L.; Peiró, J.; Franke, V., Computational modelling of 1d blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Int J Numer Methods Fluids, 43, 6-7, 673-700 (2003) · Zbl 1032.76729 |
| [29] | Sherwin, S.; Franke, V.; Peiró, J.; Parker, K., One-dimensional modelling of a vascular network in space-time variables, J Eng Math, 47, 3, 217-250 (2003) · Zbl 1200.76230 |
| [30] | Shi, Y.; Lawford, P. V.; Hose, R., Review of Zero-D and 1-D models of blood flow in the cardiovascular system, Biomed Eng Online, 10, 1, 33 (2011) |
| [31] | Wang, Z.; Li, G.; Delestre, O., Well-balanced finite difference weighted essentially non-oscillatory schemes for the blood flow model, Int J Numer Methods Fluids, 82, 9, 607-622 (2016) |
| [32] | Xiao, N.; Alastruey, J.; Alberto Figueroa, C., A systematic comparison between 1-d and 3-d hemodynamics in compliant arterial models, Int J Numer Method Biomed Eng, 30, 2, 204-231 (2014) |
| [33] | Xing, Y., Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium, J Comput Phys, 257, 536-553 (2014) · Zbl 1349.76289 |
| [34] | Xing, Y.; Shu, C.-W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J Comput Phys, 208, 206-227 (2005) · Zbl 1114.76340 |
| [35] | Xing, Y.; Shu, C.-W., A new approach of high order well-balanced finite volume WENO schemes and discontinuous galerkin methods for a class of hyperbolic systems with source terms, Commun Comput Phys, 1, 100-134 (2006) · Zbl 1115.65096 |
| [36] | Xing, Y.; Shu, C.-W., High order well-balanced finite volume WENO schemes and discontinuous galerkin methods for a class of hyperbolic systems with source terms, J Comput Phys, 214, 567-598 (2006) · Zbl 1089.65091 |
| [37] | Xing, Y.; Shu, C.-W., High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J Sci Comput, 54, 645-662 (2013) · Zbl 1260.76022 |
| [38] | Xing, Y.; Shu, C.-W., A survey of high order schemes for the shallow water equations, J Math Study, 47, 221-249 (2014) · Zbl 1324.76035 |
| [39] | Xing, Y.; Zhang, X.; Shu, C.-W., Positivity-preserving high order well-balanced discontinuous galerkin methods for the shallow water equations, Adv Water Resour, 33, 1476-1493 (2010) |
| [40] | Young, T., Hydraulic investigations, subservient to an intended croonian lecture on the motion of the blood, Philos Trans R Soc Lond, 98, 164-186 (1808) |
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