A 'well-balanced' finite volume scheme for blood flow simulation

International Journal for Numerical Methods in Fluids

Volumn 72, Issue 2, 2013, Pages 177-205

  Delestre, O ^a,b   Lagree P Y ^a  

^a SORBONNE UNIVERSITÉ   (France)

^b UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

Author keywords

Blood flow simulation; Finite volume scheme; Hydrostatic reconstruction; Man at eternal rest; Semi analytical solutions; Shallow water; Well balanced scheme

Indexed keywords

BLOOD FLOW SIMULATIONS; FINITE VOLUME SCHEMES; MAN AT ETERNAL REST; SEMI-ANALYTICAL SOLUTION; SHALLOW WATERS; WELL- BALANCED SCHEMES;

EQUATIONS OF MOTION; FLOW SIMULATION;

HEMODYNAMICS;

EID: 84876330203     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.3736     Document Type: Article

References (88)

1. Xiu D, Sherwin SJ. Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network. Journal of Computational Physics 2007; 226:1385-1407.
2. Parker KH. A brief history of arterial wave mechanics. Medical & Biological Engineering & Computing Feb 2009; 47:111-118.
3. Lighthill J. Waves in Fluids. Cambridge University Press: Cambridge, 1978.
4. Pedley TJ. The Fluid Mechanics of Large Blood Vessel. Cambridge Monographs on Mechanics, 1980.
5. Van Steenhoven AA, Van Dongen MEH. Model studies of the aortic pressure rise just after valve closure. Journal of Fluid Mechanics 1986; 166:93-113.
6. Zagzoule M, Marc-Vergnes JP. A global mathematical model of the cerebral circulation in man. Journal of Biomechanics 1986; 19(12):1015-1022.
7. Stergiopulos N, Young DF, Rogge TR. Computer simulation of arterial flow with applications to arterial and aortic stenoses. Journal of Biomechanics 1992; 25(12):1477-1488.
8. Olufsen MS. Structured tree outflow condition for blood flow in larger systemic arteries. American Journal of Physiology - Heart and Circulatory Physiology 1999; 276:257-268.
9. Lagrée PY, Rossi M. Etude de l'écoulement du sang dans les artères: effets nonlinéaires et dissipatifs. Comptes Rendus de l'Académie des Sciences, Paris 1996:401-408.
10. Lagrée PY. An inverse technique to deduce the elasticity of a large artery. The European Physical Journal 2000; 9:153-163.
11. Sherwin SJ, 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. International Journal for Numerical Methods in Fluids 2003; 43:673-700. DOI: 10.1002/fld.543.
12. Fernández MA, Milišić V, Quarteroni A. Analysis of a geometrical multiscale blood flow model based on the coupling of ODE's and hyperbolic PDE's. Technical Report 5127, INRIA, Feb 2004.
13. Wibmer M. One-dimensional simulation of arterial blood flow with applications. PhD Thesis, eingereicht an der Technischen Universitat Wien, Fakultat fur Technische Naturwissenschaften und Informatik, Jan 2004.
14. Martin V, Clément F, Decoene A, Gerbeau JF. Parameter identification for a one-dimensional blood flow model. In ESAIM: PROCEEDINGS, Vol.14, Gerbeau ECJF (ed), 2005; 174-200. DOI: 10.1051/proc:2005014. http://www.esaim-proc.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/proc/abs/2005/01/martin/martin.html.
15. Formaggia L, Lamponi D, Tuveri M, Veneziani A. Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Computer Methods in Biomechanics and Biomedical Engineering 2006; 9:273-288.
16. Cavallini N, Caleffi V, Coscia V. Finite volume and WENO scheme in one-dimensional vascular system modelling. Computers and Mathematics with Applications 2008; 56:2382-2397. DOI: 10.1016/J.camwa.2008.05.039.
17. Alastruey J, Parker KH, Peiró J, Sherwin SJ. Lumped parameter outflow models for 1D blood flow simulations: effect on pulse waves and parameter estimation. Communications in Computational Physics 2008; 4:317-336.
18. Cavallini N, 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; 147-170. (Available from: http://dx.doi.org/10.1007/978-3-642-04068-9_9).
19. Willemet M, Lacroix V, Marchandise E. Inlet boundary conditions for blood flow simulations in truncated arterial networks. Journal of Biomechanics 2011; 44(5):897-903. DOI: 10.1016/j.jbiomech.2010.11.036. (Available from: http://www.sciencedirect.com/science/article/B6T82-51V8Y64-1/2/eb86e3d4873bd8ec019a622d0c33ca36).
20. Marchandise E, Chevaugeon N, Remacle JF. Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems. Journal of Computational and Applied Mathematics 2008; 215(2):484-494. DOI: 10.1016/j.cam.2006.03.061. (Available from: http://www.sciencedirect.com/science/article/B6TYH-4MN3V8R-3/2/0bb0cd0d831f13f6105135c1f6c77587).Proceedings of the Third International Conference on Advanced Computational Methods in Engineering (ACOMEN 2005).
21. Fernández MA, Gerbeau JF, Grandmont C. A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. International Journal for Numerical Methods in Engineering 2007; 69(4):794-821. DOI: 10.1002/nme.1792.
22. van de Vosse F, de Hart J, van Oijen C, Bessems D. Finite-element-based computational methods for cardiovascular fluid-structure interaction. Journal of Engineering Mathematics 2003; 47:335-368.
23. Saito M, Ikenaga Y, Matsukawa M, Watanabe Y, Asada T, Lagrée PY. One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental. Journal of Biomechanical Engineering Dec 2011; 133:121005-1-121005-9. DOI: 10.1115/1.4005472.
24. Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Annals of Biomedical Engineering 2000; 28:1281-1299.
25. Fullana JM, Zaleski S. A branched one-dimensional model of vessel networks. Journal of Fluid Mechanics 2009; 621:183-204.
26. Pindera MZ, Ding H, Athavale MM, Chen Z. Accuracy of 1D microvascular flow models in the limit of low Reynolds numbers. Microvascular Research 2009; 77(3):273-280. DOI: 10.1016/j.mvr.2008.11.006. (Available from: http://www.sciencedirect.com/science/article/B6WN8-4V59W3D-1/2/34024497531a550ed28ec3a5aeefc401).
27. de Saint Venant AJC. Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit. Comptes Rendus de l'Académie des Sciences 1871; 73:147-154.
28. Goutal N, Maurel F. A finite volume solver for 1D shallow-water equations applied to an actual river. International Journal for Numerical Methods in Fluids 2002; 38:1-19. DOI: 10.1002/fld.201.
29. Burguete J, García-Navarro P, Murillo J. Friction term discretization and limitation to preserve stability and conservation in the 1D shallow-water model: application to unsteady irrigation and river flow. International Journal for Numerical Methods in Fluids 2008; 58(4):403-425. DOI: 10.1002/fld.1727.
30. Delestre O, Cordier S, James F, Darboux F. Simulation of rain-water overland-flow. Proceedings of the 12th International Conference on Hyperbolic Problems, University of Maryland, College Park (USA), 2008. E. Tadmor, J.-G. Liu and A. Tzavaras (eds)., Proceedings of Symposia in Applied Mathematics 67, Amer. Math. Soc., 537-546, 2009.
31. Delestre O, James F. Simulation of rainfall events and overland flow. Proceedings of X International Conference Zaragoza-Pau on Applied Mathematics and Statistics, Jaca, Spain, 2009. september 2008, Monografías Matemáticas García de Galdeano.
32. Tatard L, Planchon O, Wainwright J, Nord G, Favis-Mortlock D, Silvera N, Ribolzi O, Esteves M, Huang CH. Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil. Journal of Hydrology Jan 2008; 348(1-2):1-12. DOI: 10.1016/j.jhydrol.2007.07.016.
33. th CADAM Workshop, Zaragoza, 1999; 63-94.
34. Valiani A, Caleffi V, Zanni A. Case study: Malpasset dam-break simulation using a two-dimensional finite volume methods. Journal of Hydraulic Engineering 2002; 128(5):460-472.
35. Sampson J, Easton A, Singh M. Modelling the effect of proposed channel deepening on the tides in Port Phillip Bay, 2005. (Available from: http://hdl.handle.net/1959.3/2405).
36. Dutykh D. Modélisation mathématique des tsunamis. PhD Thesis, École normale supérieure de Cachan.
37. Popinet S. Quadtree-adaptative tsunami modelling. Ocean Dynamics May 2011; 61(9):1261-1285. DOI: 10.1007/s10236-011-0438-z.
38. Bermudez A, Vazquez ME. Upwind methods for hyperbolic conservation laws with source terms. Computers & Fluids 1994; 23(8):1049-1071. DOI: 10.1016/0045-7930(94)90004-3. (Available from: http://www.sciencedirect.com/science/article/B6V26-47YMXJT-16/2/4485b4adb616198b768daff172258660).
39. Bermúdez A, Dervieux A, Desideri JA, Vázquez ME. Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Computer Methods in Applied Mechanics and Engineering 1998; 155(1-2):49-72. DOI: 10.1016/S0045-7825(97)85625-3. (Available from: http://www.sciencedirect.com/science/article/B6V29-3WN739F-4/2/3120ee07ea5c43b3318fbfb6c05d2dde).
40. LeVeque RJ. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. Journal of Computational Physics 1998; 146(1):346-365. DOI: 10.1006/1998.6058jcph. (Available from: http://www.sciencedirect.com/science/article/B6WHY-45J58TW-22/2/72a8af8c9e23f63b0df9484475f7e2df).
41. Jin S. A steady-state capturing method for hyperbolic systems with geometrical source terms. M2AN 2001; 35(4):631-645. DOI: 10.1051/m2an:2001130. (Available from: http://dx.doi.org/10.1051/m2an:2001130).
42. Greenberg JM, LeRoux A. A well-balanced scheme for the numerical processing of source terms in hyperbolic equation. SIAM Journal on Numerical Analysis 1996; 33:1-16.
43. Gosse L. A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Computers & Mathematics with Applications 2000; 39(9-10):135-159. DOI: 10.1016/S0898-1221(00)00093-6. (Available from: http://www.sciencedirect.com/science/article/B6TYJ-40NFTJN-D/2/320458e48d7fbefdf910aaf251d7255c).
44. Gallouët T, Hérard JM, Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Computers & Fluids 2003; 32:479-513.
45. Kurganov A, Levy D. Central-upwind schemes for the Saint-Venant system. Mathematical Modelling and Numerical Analysis 2002; 36:397-425.
46. Perthame B, Simeoni C. A kinetic scheme for the Saint-Venant system with a source term. Calcolo 2001; 38:201-231. DOI: 10.1007/s10092-001-8181-3. (Available from: http://dx.doi.org/10.1007/s10092-001-8181-3).
47. Audusse E, Bouchut F, Bristeau MO, Klein R, Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing 2004; 25(6):2050-2065. DOI: 10.1137/S1064827503431090.
48. Bouchut F. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Vol.2/2004. Birkhäuser: Basel, 2004. DOI: 10.1007/b95203.
49. Vàzquez-Cendòn ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. Journal of Computational Physics 1999; 148:497-526.
50. Chinnayya A, LeRoux AY. A new General Riemann Solver for the Shallow Water equations, with friction and topography, 1999. (Available from: http://www.math.ntnu.no/conservation/1999/021.html),Preprint.
51. Lukáčová-Medvid'ová M, Vlk Z. Well-balanced finite volume evolution Galerkin methods for the shallow water equations with source terms. International Journal for Numerical Methods in Fluids 2005; 47(10-11):1165-1171.
52. Lukáčová-Medvid'ová M, Teschke U. Comparison study of some finite volume and finite element methods for the shallow water equations with bottom topography and friction terms. Journal of Applied Mathematics and Mechanics ZAMM 2006; 86(11):874-891. DOI: 10.1002/zamm.200510280.
53. Lee SH, Wright NG. Simple and efficient solution of the shallow water equations with source terms. International Journal for Numerical Methods in Fluids 2010; 63:313-340. DOI: 10.1002/fld.2071.
54. Hou TY, LeFloch PG. Why nonconservative schemes converge to wrong solutions: error analysis. Mathematics of Computation Apr 1994; 62(206):497-530.
55. Berthon C, Coquel F. Nonlinear projection methods for multi-entropies Navier-Stokes systems. Mathematics of Computation 2007; 76:1163-1194.
56. Castro MJ, LeFloch PG, Luz Muñoz Ruiz M, Parés C. Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. Journal of Computational Physics 2008; 227(17):8107-8129. DOI: 10.1016/j.jcp.2008.05.012. (Available from: http://www.sciencedirect.com/science/article/B6WHY-4SMNY18-1/2/a96e0bdf121ac34604742fdc9165bd07).
57. Lagrée PY, Lorthois S.The RNS/PRANDTl equations and their link with other asymptotic descriptions. Application to the computation of the maximum value of the wall shear stress in a pipe. International Journal of Engineering Science 2005; 43(3-4):352-378.
58. Zagzoule M, Khalid-Naciri J, Mauss J. Unsteady wall shear stress in a distensible tube. Journal of Biomechanics 1991; 24(6):435-439.
59. Kundu PK, Cohen IM, Hu HH, Third Edition, Academic Press, 2004.
60. Kundu PK, Cohen IM, Fourth Edition, Academic Press, Elsevier, The Boulevard, Langford Lane Kidlington: Oxford, OX5 1GB, UK, 2008.
61. Castro MJ, Pardo M A, Parés C. Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Mathematical Models and Methods in Applied Sciences 2007; 17(12):2065-2113.
62. Noelle S, Xing Y, Shu CW. High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. Journal of Computational Physics 2007; 226:29-58. DOI: 10.1016/j.jcp.2007.03.031. (Available from: http://www.sciencedirect.com/science/article/B6WHY-4NG3TH4-3/2/546ff3d247ae9db8ed41e3de935ee892).
63. Thanh MD, Fazlul Karim M, Ismail AIM. Well-balanced scheme for shallow water equations with arbitrary topography. International Journal of Dynamical Systems and Differential Equations 2008; 1(3):196-204.
64. Bouchut F, Morales de Luna T. A Subsonic-Well-Balanced Reconstruction Scheme for Shallow Water Flows. SIAM Journal on Numerical Analysis 2010; 48:1733-1758.
65. Fiedler RF, Ramirez JA. A numerical method for simulating discontinuous shallow flow over an infiltrating surface. International Journal for Numerical Methods in Fluids 2000; 32:219-240.
66. Bristeau MO, Coussin B. Boundary conditions for the shallow water equations solved by kinetic schemes. Technical Report 4282, INRIA, Oct 2001.
67. Liang Q, Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms. Advances in Water Resources 2009; 32(6):873-884. DOI: 10.1016/j.advwatres.2009.02.010. (Available from: http://www.sciencedirect.com/science/article/B6VCF-4VR9FHT-3/2/5218868dcfded7d34380666f23000855).
68. Mangeney-Castelnau A, Bouchut F, Vilotte JP, Lajeunesse E, Aubertin A, Pirulli M. On the use of Saint-Venant equations to simulate the spreading of a granular mass. Journal of Geophysical Research 2005; 110:1-17.
69. Mangeney A, Bouchut F, Thomas N, Vilotte JP, Bristeau MO. Numerical modeling of self-channeling granular flow and of their levee-channel deposits. Journal of Geophysical Research May 2007; 112(F2):1-21. DOI: 10.1029/2006JF000469.
70. Harten A, Lax PD, van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review Jan 1983; 25(1):35-61.
71. Marche F, Bonneton P, Fabrie P, Seguin N. Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes. International Journal for Numerical Methods in Fluids 2007; 53:867-894. DOI: 10.1002/fld.1311.
72. Marche F, Berthon C. A positive preserving high order VFRoe scheme for shallow water equations: A class of relaxation schemes. SIAM Journal on Scientific Computing Aug 2008; 30(5):2587-2612. (Available from: http://hal.archives-ouvertes.fr/hal-00370486/en/),75M12, 35L65, 65M12.
73. Audusse E. Schémas cinétiques pour le système de Saint-Venant. Master's Thesis, Université Paris 6, 1999.
74. Audusse E, Bristeau MO. A well-balanced positivity preserving "second-order" scheme for shallow water flows on unstructured meshes. Journal of Computational Physics 2005; 206:311-333. DOI: 10.1016/j.jcp.2004.12.016.
75. Delestre O. Simulation du ruissellement d'eau de pluie sur des surfaces agricoles/ rain water overland flow on agricultural fields simulation. PhD Thesis, Université d'Orléans (in French). available from TEL: tel.archives-ouvertes.fr/INSMI/tel-00531377/fr, Jul 2010.
76. LeVeque RJ. Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press: Cambridge, 2002.
77. van Leer B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. Journal of Computational Physics 1979; 32(1):101-136. DOI: 10.1016/0021-9991(79)90145-1. (Available from: http://www.sciencedirect.com/science/article/B6WHY-4DD1N8T-C5/2/9b051d1cfcff715a3d0f4b7b7b0397cc).
78. Delestre O, Marche F. A numerical scheme for a viscous shallow water model with friction. Journal of Scientific Computing 2010. DOI: 10.1007/s10915-010-9393-y.
79. Harten A, Osher S, Engquist B, Chakravarthy SR. Some results on uniformly high-order accurate essentially nonoscillatory schemes. Applied Numerical Mathematics 1986; 2(3-5):347-377. DOI: 10.1016/0168-9274(86)90039-5. (Available from: http://www.sciencedirect.com/science/article/B6TYD-45GVV8T-J/2/830ca2dce5e3fb34ace9fa53ae5ab76b), special Issue in Honor of Milt Rose's Sixtieth Birthday.
80. Harten A, Engquist B, Osher S, Chakravarthy SR. Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of Computational Physics Aug 1987; 71:231-303.
81. Shu CW, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 1988; 77(2):439-471. DOI: 10.1016/0021-9991(88)90177-5. (Available from: http://www.sciencedirect.com/science/article/B6WHY-4DD1T6W-MM/2/bef99e4b67bd7132c8af1984c34ce57e).
82. Bouchut F, Ounaissa H, Perthame B. Upwinding of the source term at interfaces for Euler equations with high friction. Computers and Mathematics with Applications 2007; 53:361-375. DOI: 10.1016/j.camwa.2006.02.055.
83. Noelle S, Pankratz N, Puppo G, Natvig JR. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. Journal of Computational Physics 2006; 213(2):474-499. DOI: 10.1016/j.jcp.2005.08.019. (Available from: ttp://www.sciencedirect.com/science/article/B6WHY-4HB4DN4-1/2/15c5962d035c22b4fee10302b2606fa6).
84. Stoker JJ. Water Waves: The Mathematical Theory with Applications. Interscience Publishers: New York, USA, 1957.
85. LeVeque RJ. Numerical Methods for Conservation Laws, Lectures in Mathematics. ETH Zurich, 1992.
86. Ockendon H, Tayler AB. Applied Mathematical Sciences, Vol.43. Springer, 1983.
87. Kirkman R, Moore T, Adlard C. The Walking Dead. Image Comics: Berkeley, 2003.
88. Womersley J.On the oscillatory motion of a viscous liquid in thin-walled elastic tube: I. Philosophical Magazine 1955; 46:199-221.