Hostname: page-component-788cddb947-r7bls Total loading time: 0 Render date: 2024-10-10T11:05:32.131Z Has data issue: false hasContentIssue false

A High Order Well-Balanced Finite Volume WENO Scheme for a Blood Flow Model in Arteries

Published online by Cambridge University Press:  31 January 2018

Zhonghua Yao*
Affiliation:
School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, PR China
Gang Li*
Affiliation:
School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, PR China
Jinmei Gao*
Affiliation:
School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, PR China
*
*Corresponding author. Email addresses:yaozhonghuavip@163.com (Z. Yao), gangli1978@163.com (G. Li), gjinmei1979@163.com (J. Gao)
*Corresponding author. Email addresses:yaozhonghuavip@163.com (Z. Yao), gangli1978@163.com (G. Li), gjinmei1979@163.com (J. Gao)
*Corresponding author. Email addresses:yaozhonghuavip@163.com (Z. Yao), gangli1978@163.com (G. Li), gjinmei1979@163.com (J. Gao)
Get access

Abstract

The numerical simulations for the blood flow in arteries by high order accurate schemes have a wide range of applications in medical engineering. The blood flow model admits the steady state solutions, in which the flux gradient is non-zero and is exactly balanced by the source term. In this paper, we present a high order finite volume weighted essentially non-oscillatory (WENO) scheme, which preserves the steady state solutions and maintains genuine high order accuracy for general solutions. The well-balanced property is obtained by a novel source term reformulation and discretisation, combined with well-balanced numerical fluxes. Extensive numerical experiments are carried out to verify well-balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.

MSC classification

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Cavallini, N., Caleffi, V. and Coscia, V., Finite volume and WENO scheme in one-dimensional vascular system modelling, Comput. Math. Appl. 56, 23822397 (2008).Google Scholar
[2] Cavallini, N. and Coscia, V., One-dimensional modelling of venous pathologies: Finite volume and WENO schemes, In Advances in Mathematical Fluid Mechanics, Rannacher R, Sequeira A (eds). Springer: Berlin Heidelberg, 2010.Google Scholar
[3] Delestre, O. and Lagrée, P.Y., A ‘well-balanced’ finite volume scheme for blood flow simulation, Int. J. Numer. Meth. Fl. 72, 177205 (2013).Google Scholar
[4] Delestre, O., Lucas, C., Ksinant, P.A., Darboux, F., Laguerre, C., V, T.N.T., James, F. and Cordier, S., SWASHES: A compilation of Shallow Water Analytic Solutions for Hydraulic and Environmental Studies, Int. J. Numer. Meth. Fl. 72, 269300 (2013).Google Scholar
[5] Formaggia, L., Lamponi, D., Tuveri, M. and Veneziani, A., Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart, Comput. Method Biomech. Biomed. Engin. 9, 273288 (2006).Google Scholar
[6] Greenberg, J.M. and LeRoux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33, 116 (1996).Google Scholar
[7] Jiang, G. and Shu, C.W., Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202228 (1996).Google Scholar
[8] Kolachalama, V.B., Bressloff, N.W., Nair, P.B. and Shearman, C.P., Predictive Haemodynamics in a one-dimensional human carotid artery bifurcation. Part I: Application to stent design, IEEE T. Bio-Med. Eng. 54, 802812 (2007).Google Scholar
[9] Murillo, J. and García-Navarro, P., A Roe type energy balanced solver for 1D arterial blood flow and transport, Comput. Fluids 117, 149167 (2015).Google Scholar
[10] Müller, L.O., C. Parés and Toro, E.F., Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties, J. Comput. Phys. 242, 5385 (2013).Google Scholar
[11] Noelle, S., Xing, Y. L. and Shu, C.W., High-order well-balanced schemes, In: Numerical Methods for Balance Laws (Puppo, G. and Russo, G. eds). Quaderni di Matematica (2010).Google Scholar
[12] Shu, C.W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, In Quarteroni, A., editor, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325432. Lecture Notes in Mathematics, volume 1697, Springer, 1998.Google Scholar
[13] Shu, C.W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77, 4394718 (1988).Google Scholar
[14] Womersley, J., On the oscillatory motion of a viscous liquid in thin-walled elastic tube: I., Phil. Mag. 46, 199221 (1955).Google Scholar
[15] Wibmer, M., One-dimensional simulation of arterial blood flow with applications, PhD Thesis, eingereicht an der Technischen UniversitatWien, Fakultat fur Technische Naturwissenschaften und Informatik, January, 2004.Google Scholar
[16] Wang, Z.Z., Li, G. and Delestre, O.. Well-balanced finite difference weighted essentially non-oscillatory schemes for the blood flow model, Int. J. Numer. Meth. Fl. 82, 607622 (2016).Google Scholar
[17] Xing, Y.L., Shu, C.W. and Noelle, S., On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations, J. Sci. Comput. 48, 339349 (2011).Google Scholar