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

  • Zhonghua Yao (a1), Gang Li (a1) and Jinmei Gao (a1)
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.

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*Corresponding author. Email addresses: yaozhonghuavip@163.com (Z. Yao), gangli1978@163.com (G. Li), gjinmei1979@163.com (J. Gao)
References
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[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).
[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.
[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).
[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).
[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).
[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).
[7] Jiang, G. and Shu, C.W., Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202228 (1996).
[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).
[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).
[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).
[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).
[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.
[13] Shu, C.W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77, 4394718 (1988).
[14] Womersley, J., On the oscillatory motion of a viscous liquid in thin-walled elastic tube: I., Phil. Mag. 46, 199221 (1955).
[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.
[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).
[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).
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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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