A semi-implicit finite volume scheme for blood flow in elastic and viscoelastic vessels. (English) Zbl 07766248
Summary: A novel staggered semi-implicit finite volume method for the simulation of one-dimensional blood flow in networks of elastic and viscoelastic vessels is proposed. The one-dimensional blood flow model is split into three subsystems: one containing the nonlinear convective terms, a second one for the viscoelastic terms and a final subsystem for the pressure-related terms. An explicit approach is then employed to discretize the convective subsystem, while the viscous and pressure terms are treated implicitly. This leads to a CFL-type time step restriction which depends only on the bulk velocity of the flow and not on the speed of the pressure waves or on the visco elasticity. As such, the method is computationally efficient in case of low speed index, which is equivalent to a low Mach regime for the Navier-Stokes equations. To extend the proposed methodology to the case of networks, a novel and very simple 3D approach for the treatment of junctions is proposed. Each junction is represented by a unique three-dimensional primal cell and the Euler equations are employed to approximate the velocity and pressure unknowns. A multidimensional numerical flux then takes into account the elementary information of the junction geometry, namely normal vectors and areas of the incident vessels, when approximating the blood flow at the cell interfaces embedded in the 3D cell. As such, the final scheme, based on guaranteeing mass and momentum conservation through the junction, is able to capture reflecting waves properly considering the effect of the different incident angles of vessels at a junction. The proposed methodology is carefully validated in the context of single elastic and viscoelastic vessels and small networks showing a good agreement with available analytical, experimental and numerical data.
MSC:
| 76Mxx | Basic methods in fluid mechanics |
| 92Cxx | Physiological, cellular and medical topics |
| 76Zxx | Biological fluid mechanics |
Keywords:
semi-implicit finite volume scheme; flux vector splitting; 1D blood flow model; 3D junction approach; viscoelastic vesselsSoftware:
HE-E1GODFReferences:
| [1] | Formaggia, L.; Quarteroni, A.; Veneziani, A., Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, vol. 1 (2010), Springer Science & Business Media |
| [2] | Quarteroni, A.; Manzoni, A.; Vergara, C., The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications, Acta Numer., 26, 365-590 (2017) · Zbl 1376.92016 |
| [3] | Taylor, C.; Draney, M.; Ku, J.; Parker, D.; Steele, B.; Wang, K.; Zarins, C., Predictive medicine: computational techniques in therapeutic decision-making, Comput. Aided Surg.: Off. J. Int. Soc. Comput. Aided Surg. (ISCAS), 4, 5, 231-247 (1999) |
| [4] | Toro, E. F., Brain venous haemodynamics, neurological diseases and mathematical modelling. A review, Appl. Math. Comput., 272, 542-579 (2016) · Zbl 1410.76487 |
| [5] | Figueroa, C.; Taylor, C.; Marsden, A., Blood flow, (Encyclopedia of Computational Mechanics (2017)), 1-31 |
| [6] | Fossan, F. E.; Sturdy, J.; Müller, L. O.; Strand, A.; Bråten, A. T.; Jørgensen, A.; Wiseth, R.; Hellevik, L. R., Uncertainty quantification and sensitivity analysis for computational FFR estimation in stable coronary artery disease, Cardiovasc. Eng. Technol., 9, 4, 597-622 (2018) |
| [7] | Quarteroni, A.; Dedé, L.; Regazzoni, F.; Vergara, C., A mathematical model of the human heart suitable to address clinical problems, Jpn. J. Ind. Appl. Math., 1-21 (2023) |
| [8] | Bazilevs, Y.; Calo, V. M.; Zhang, Y.; Hughes, T., Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Comput. Mech., 38, 310-322 (2006) · Zbl 1161.74020 |
| [9] | Crosetto, P.; Reymond, P.; Deparis, S.; Kontaxakis, D.; Stergiopulos, N.; Quarteroni, A., Fluid-structure interaction simulation of aortic blood flow, Comput. Fluids, 43, 1, 46-57 (2011) |
| [10] | Lucca, A.; Busto, S.; Dumbser, M., An implicit staggered hybrid finite volume/finite element solver for the incompressible Navier-Stokes equations, East Asian J. Appl. Math., 13, 671-716 (2023) · Zbl 1527.65079 |
| [11] | Grinberg, L.; Cheever, E.; Anor, T.; Madsen, J.; Karniadakis, G., Modeling blood flow circulation in intracranial arterial networks: a comparative 3D/1D simulation study, Ann. Biomed. Eng., 39, 297-309 (2011) |
| [12] | Xiao, N.; Alastruey, J.; Figueroa, C. A., A systematic comparison between 1-D and 3-D hemodynamics in compliant arterial models, Int. J. Numer. Methods Biomed. Eng., 30, 2, 204 (2014) |
| [13] | Boileau, E.; Nithiarasu, P.; Blanco, P. J.; Müller, L. O.; Fossan, F. E.; Hellevik, L. R.; Donders, W. P.; Huberts, W.; Willemet, M.; Alastruey, J., A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling, Int. J. Numer. Methods Biomed. Eng., 31, 10, Article e02732 pp. (2015) |
| [14] | Matthys, K. S.; Alastruey, J.; Peiró, J.; Khir, A. W.; Segers, P.; Verdonck, P. R.; Parker, K. H.; Sherwin, S. J., Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements, J. Biomech., 40, 15, 3476-3486 (2007) |
| [15] | Bessems, D.; Giannopapa, C.; Rutten, M.; van de Vosse, F., Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels, J. Biomech., 41, 284-291 (2008) |
| [16] | Olufsen, M.; Peskin, C.; Kim, W.; Pedersen, E.; Nadim, A.; Larsen, J., Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Ann. Biomed. Eng., 28, 1281-1299 (2000) |
| [17] | Reymond, P.; Merenda, F.; Perren, F.; Rüfenacht, D.; Stergiopulos, N., Validation of a one-dimensional model of the systemic arterial tree, Am. J. Physiol., Heart Circ. Physiol., 297, H208-H222 (2009) |
| [18] | Stettler, J.; Niederer, P.; Anliker, M.; Casty, M., Theoretical analysis of arterial hemodynamics including the influence of bifurcations. Part II: Critical evaluation of theoretical model and comparison with noninvasive measurements of flow patterns in normal and pathological cases, Ann. Biomed. Eng., 9, 165-175 (1981) |
| [19] | Avolio, A. P., Multi-branched model of the human arterial system, Med. Biol. Eng. Comput., 18, 709-718 (1980) |
| [20] | O’Rourke, M. F.; Avolio, A. P., Pulsatile flow and pressure in human systemic arteries: studies in man and in a multibranched model of the human systemic arterial tree, Circ. Res., 46, 363-372 (1980) |
| [21] | Parker, K. H.; Jones, C. J.H., Forward and backward running waves in the arteries: analysis using the method of characteristics, J. Biomech. Eng., 112, 322-326 (1990) |
| [22] | Jin, W.; Alastruey, J., Arterial pulse wave propagation across stenoses and aneurysms: assessment of one-dimensional simulations against three-dimensional simulations and in vitro measurements, J. R. Soc. Interface, 18, 177, Article 20200881 pp. (2021) |
| [23] | Steele, B.; Wan, J.; Ku, J.; Hughes, T.; Taylor, C., In vivo validation of a one-dimensional finite-element method for predicting blood flow in cardiovascular bypass grafts, IEEE Trans. Biomed. Eng., 50, 6, 649-656 (2003) |
| [24] | Müller, L. O.; Toro, E. F.; Haacke, M. E.; Utriainen, D., Impact of CCSVI on cerebral haemodynamics: a mathematical study using MRI angiographic and flow data, Phlebology: J. Venous Dis., 31, 5, 305-324 (2016) |
| [25] | Liang, F.; Fukasaku, K.; Liu, H.; Takagi, S., A computational model study of the influence of the anatomy of the circle of Willis on cerebral hyperperfusion following carotid artery surgery, Biomed. Eng. Online, 10, 1, 84 (2011) |
| [26] | Liang, F.; Takagi, S.; Himeno, R.; Liu, H., Biomechanical characterization of ventricular-arterial coupling during aging: a multi-scale model study, J. Biomech., 42, 6, 692-704 (2009) |
| [27] | Mynard, J. P.; Smolich, J. J., One-dimensional haemodynamic modeling and wave dynamics in the entire adult circulation, Ann. Biomed. Eng., 43, 6, 1443-1460 (2015) |
| [28] | Blanco, P. J.; Watanabe, S. M.; Passos, M. A.; Lemos, P. A.; Feijóo, R. A., An anatomically detailed arterial network model for one-dimensional computational hemodynamics, IEEE Trans. Biomed. Eng., 62, 2, 736-753 (2014) |
| [29] | Müller, L. O.; Toro, E. F., A global multiscale mathematical model for the human circulation with emphasis on the venous system, Int. J. Numer. Methods Biomed. Eng., 30, 7, 681-725 (2014) |
| [30] | Toro, E. F.; Celant, M.; Zhang, Q.; Contarino, C.; Agarwal, N.; Linninger, A.; Müller, L. O., Cerebrospinal fluid dynamics coupled to the global circulation in holistic setting: mathematical models, numerical methods and applications, Int. J. Numer. Methods Biomed. Eng., 38, 1 (2022) |
| [31] | Müller, L. O.; Watanabe, S. M.; Toro, E. F.; Feijoo, R. A.; Blanco, P. J., An anatomically detailed arterial-venous network model. Cerebral and coronary circulation, Front. Physiol. Comput. Physiol. Med., 14 (2023) |
| [32] | Formaggia, L.; Quarteroni, A.; Vergara, C., On the physical consistency between three-dimensional and one-dimensional models in haemodynamics, J. Comput. Phys., 244, 97-112 (2013) · Zbl 1377.76042 |
| [33] | Blanco, P. J.; Feijóo, R. A., A dimensionally-heterogeneous closed-loop model for the cardiovascular system and its applications, Med. Eng. Phys., 35, 5, 652-667 (2013) |
| [34] | Blanco, P. J.; Ares, G. D.; Urquiza, S. A.; Feijóo, R. A., On the effect of preload and pre-stretch on hemodynamic simulations: an integrative approach, Biomech. Model. Mechanobiol., 15, 3, 593-627 (2016) |
| [35] | Formaggia, L.; Lamponi, D.; Quarteroni, A., One-dimensional models for blood flow in arteries, J. Eng. Math., 47, 251-276 (2003) · Zbl 1070.76059 |
| [36] | Bertaglia, G.; Caleffi, V.; Valiani, A., Modeling blood flow in viscoelastic vessels: the 1D augmented fluid-structure interaction system, Comput. Methods Appl. Mech. Eng., 360, Article 112772 pp. (2020) · Zbl 1441.76146 |
| [37] | Toro, E. F.; Siviglia, A., Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions, Commun. Comput. Phys., 13, 2, 361-385 (2013) · Zbl 1373.76362 |
| [38] | Spilimbergo, A.; Toro, E. F.; Müller, L. O., One-dimensional blood flow with discontinuous properties and transport: mathematical analysis and numerical schemes, Commun. Comput. Phys., 29, 3, 649-697 (2021) · Zbl 1480.76171 |
| [39] | Bertaglia, G.; Navas-Montilla, A.; Valiani, A.; Monge-García, M.; Murillo, J.; Caleffi, V., Computational hemodynamics in arteries with the one-dimensional augmented fluid-structure interaction system: viscoelastic parameters estimation and comparison with in-vivo data, J. Biomech., 100, Article 109595 pp. (2020) |
| [40] | Lambert, J. W., On the nonlinearities of fluid flow in nonrigid tubes, J. Franklin Inst., 266, 2, 83-102 (1958) |
| [41] | Barnard, A. L.; Hunt, W.; Timlake, W.; Varley, E., A theory of fluid flow in compliant tubes, Biophys. J., 6, 6, 717-724 (1966) |
| [42] | Hughes, T. J.; Lubliner, J., On the one-dimensional theory of blood flow in the large vessels, Math. Biosci., 18, 161-170 (1973) · Zbl 0262.92004 |
| [43] | Euler, L., Principia pro motu sanguinis per arterias determinando, Opera postuma, 814-823 (1862) |
| [44] | Young, T., I. The Croonian lecture. On the functions of the heart and arteries, Philos. Trans. R. Soc. Lond., 99, 1-31 (1809) |
| [45] | Hellevik, L. R.; Vierendeels, J.; Kiserud, T.; Stergiopulos, N.; Irgens, F.; Dick, E.; Riemslagh, K.; Verdonck, P., An assessment of ductus venosus tapering and wave transmission from the fetal heart, Biomech. Model. Mechanobiol., 8, 6, 509-517 (2009) |
| [46] | Brook, B. S.; Falle, S. A.E. G.; Pedley, T. J., Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state, J. Fluid Mech., 396, 223-256 (1999) · Zbl 0971.76052 |
| [47] | Müller, L. O.; Montecinos, G. I.; Toro, E. F., Some issues in modelling venous haemodynamics, (Vázquez-Cendón, P. G.-N. M.E.; Hidalgo, A.; Cea, L., Numerical Methods for Hyperbolic Equations: Theory and Applications. An International Conference to Honour Professor E.F. Toro (2013), Taylor & Francis Group: Taylor & Francis Group Boca Raton, London, New York, Leiden), 347-354 |
| [48] | Sherwin, S. J.; Formaggia, L.; Peirò, 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, 673-700 (2003) · Zbl 1032.76729 |
| [49] | Ghitti, B.; Berthon, C.; Le, M.; Toro, E., A fully well-balanced scheme for the 1d blood flow equations with friction source term, J. Comput. Phys., 421, Article 109750 pp. (2020) · Zbl 07508369 |
| [50] | Pimentel-García, E.; Müller, L. O.; Toro, E. F.; Parés, C., High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties, J. Comput. Phys., 475, Article 111869 pp. (2023) · Zbl 07649287 |
| [51] | Bermúdez, A.; Busto, S.; Dumbser, M.; Ferrín, J.; Saavedra, L.; Vázquez-Cendón, M., A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows, J. Comput. Phys., 421, Article 109743 pp. (2020) · Zbl 07508366 |
| [52] | Luo, H.; Baum, J. D.; Löhner, R., A fast, matrix-free implicit method for compressible flows on unstructured grids, J. Comput. Phys., 146, 2, 664-690 (1998) · Zbl 0931.76045 |
| [53] | Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 230, 1147-1170 (2011) · Zbl 1391.76353 |
| [54] | Carson, J.; Van Loon, R., An implicit solver for 1d arterial network models, Int. J. Numer. Methods Biomed. Eng., 33, 7, Article e2837 pp. (2017) |
| [55] | Liu, M.; Gao, G.; Zhu, H.; Jiang, C., A cell-based smoothed finite element method stabilized by implicit SUPG/SPGP/fractional step method for incompressible flow, Eng. Anal. Bound. Elem., 124, 194-210 (2021) · Zbl 1464.76056 |
| [56] | Setzwein, F.; Ess, P.; Gerlinger, P., An implicit high-order k-exact finite-volume approach on vertex-centered unstructured grids for incompressible flows, J. Comput. Phys., 446, Article 110629 pp. (2021) · Zbl 07516451 |
| [57] | Park, J.; Munz, C., Multiple pressure variables methods for fluid flow at all Mach numbers, Int. J. Numer. Methods Fluids, 49, 8, 905-931 (2005) · Zbl 1170.76342 |
| [58] | Toro, E. F.; Vázquez-Cendón, M. E., Flux splitting schemes for the Euler equations, Comput. Fluids, 70, 1-12 (2012) · Zbl 1365.76243 |
| [59] | Noelle, S.; Bispen, G.; Arun, K. R.; Lukáčová-Medvid’ová, M.; Munz, C. D., An asymptotic preserving all Mach number scheme for the Euler equations of gas dynamics (2014), Institut für Geometrie und Praktische Mathematik, RWTH Aachen: Institut für Geometrie und Praktische Mathematik, RWTH Aachen Germany, Report 348 · Zbl 1321.76053 |
| [60] | Tavelli, M.; Dumbser, M., A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers, J. Comput. Phys., 341, 341-376 (2017) · Zbl 1376.76028 |
| [61] | Thomann, A.; Puppo, G.; Klingenber, C., An All Speed Second Order IMEX Relaxation Schemefor the Euler Equations (2020) · Zbl 1480.76083 |
| [62] | Busto, S.; Río-Martín, L.; Vázquez-Cendón, M.; Dumbser, M., A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes, Appl. Math. Comput., 402, Article 126117 pp. (2021) · Zbl 1510.76082 |
| [63] | Casulli, V.; Dumbser, M.; Toro, E. F., Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems, Int. J. Numer. Methods Biomed. Eng., 28, 2, 257-272 (2012) · Zbl 1242.92019 |
| [64] | Malossi, A.; Blanco, P.; Crosetto, P.; Deparis, S.; Quarteroni, A., Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels, Multiscale Model. Simul., 11, 2, 474-506 (2013) · Zbl 1310.92017 |
| [65] | Tavelli, M.; Dumbser, M.; Casulli, V., High resolution methods for scalar transport problems in compliant systems of arteries, Appl. Numer. Math., 74, 62-82 (2013) · Zbl 1302.76134 |
| [66] | Fambri, F.; Dumbser, M.; Casulli, V., An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels, Int. J. Numer. Methods Biomed. Eng., 30, 11, 1170-1198 (2014) |
| [67] | Dumbser, M.; Iben, U.; Ioriatti, M., An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes, Appl. Numer. Math., 89, 24-44 (2015) · Zbl 1326.76070 |
| [68] | Ioriatti, M.; Dumbser, M.; Iben, U., A comparison of explicit and semi-implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime, Z. Angew. Math. Mech., 97, 11, 1358-1380 (2017) · Zbl 07776755 |
| [69] | Borsche, R., Numerical schemes for networks of hyperbolic conservation laws, Appl. Numer. Math., 108, 157-170 (2016) · Zbl 1346.65040 |
| [70] | Müller, L. O.; Leugering, G.; Blanco, P. J., Consistent treatment of viscoelastic effects at junctions in one-dimensional blood flow models, J. Comput. Phys., 314, 167-193 (2016) · Zbl 1349.76369 |
| [71] | Bellamoli, F.; Müller, L. O.; Toro, E. F., A numerical method for junctions in networks of shallow-water channels, Appl. Math. Comput., 337, 190-213 (2018) · Zbl 1427.76156 |
| [72] | Alastruey, J.; Parker, K. H.; Peiró, J.; Sherwin, S. J., Lumped parameter outflow models for 1-D blood flow simulations: effect on pulse waves and parameter estimation, Commun. Comput. Phys., 4, 317-336 (2008) · Zbl 1364.76248 |
| [73] | Bermúdez, A.; López, X.; Vázquez-Cendón, M. E., Treating network junctions in finite volume solution of transient gas flow models, J. Comput. Phys., 344, Supplement C, 187-209 (2017) · Zbl 1380.76050 |
| [74] | Liu, X.; Chertock, A.; Kurganov, A.; Wolfkill, K., One-dimensional/two-dimensional coupling approach with quadrilateral confluence region for modeling river systems, J. Sci. Comput., 81, 1297-1328 (2019) · Zbl 1429.86003 |
| [75] | Briani, M.; Puppo, G.; Ribot, M., Angle dependence in coupling conditions for shallow water equations at channel junctions, Comput. Math. Appl., 108, 49-65 (2022) · Zbl 1524.35380 |
| [76] | Murillo, J.; García-Navarro, P., A solution of the junction Riemann problem for 1D hyperbolic balance laws in networks including supersonic flow conditions on elastic collapsible tubes, Symmetry, 13, 9, 1658 (2021) |
| [77] | Piccioli, F.; Bertaglia, G.; Valiani, A.; Caleffi, V., Modeling blood flow in networks of viscoelastic vessels with the 1-D augmented fluid-structure interaction system, J. Comput. Phys., 464, Article 111364 pp. (2022) · Zbl 07540379 |
| [78] | Colombo, R. M.; Garavello, M., On the 1D modeling of fluid flowing through a junction, Discrete Contin. Dyn. Syst., Ser. B, 25, 10 (2020) · Zbl 1452.35138 |
| [79] | Formaggia, L.; Perktold, K.; Quarteroni, A., Basic mathematical models and motivations, Cardiovasc. Math.: Model. Simul. Circ. Syst., 47-75 (2009) |
| [80] | Mynard, J. P.; Nithiarasu, P., A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method, Commun. Numer. Methods Eng., 24, 5, 367-417 (2008) · Zbl 1137.92009 |
| [81] | Ghitti, B.; Toro, E. F.; Müller, L. O., Nonlinear lumped-parameter models for blood flow simulations in networks of vessels, ESAIM: Math. Model. Numer. Anal., 56, 5, 1579-1627 (2022) · Zbl 1498.76121 |
| [82] | Formaggia, L.; Veneziani, A., Reduced and Multiscale Models for the Human Cardiovascular System, Lecture Notes VKI Lecture Series, vol. 7 (2003) |
| [83] | Mynard, J. P.; Valen-Sendstad, K., A unified method for estimating pressure losses at vascular junctions, Int. J. Numer. Methods Biomed. Eng., 31, 7 (2015) |
| [84] | Colombo, R. M.; Herty, M.; Sachers, V., On 2×2 conservation laws at a junction, SIAM J. Math. Anal., 40, 2, 605-622 (2008) · Zbl 1171.35430 |
| [85] | Fullana, J.-M.; Zaleski, S., A branched one-dimensional model of vessel networks, J. Fluid Mech., 621, 183-204 (2009) · Zbl 1171.76475 |
| [86] | Wayand, I., A Staggered Semi-Implicit Discontinuous Galerkin Scheme for the Simulation of Incompressible Flows Through Junctions in Pipe Networks (May 2017) |
| [87] | Quarteroni, A.; Formaggia, L., Mathematical modelling and numerical simulation of the cardiovascular system, Handb. Numer. Anal., 12, 3-127 (2004) |
| [88] | Flaherty, J. E.; Keller, J. B.; Rubinow, S. I., Post buckling behavior of elastic tubes and rings with opposite sides in contact, SIAM J. Appl. Math., 23, 4, 446-455 (1972) · Zbl 0249.73046 |
| [89] | Ducros, F.; Ferrand, V.; Nicoud, F.; Weber, C.; Darracq, D.; Gacherieu, C.; Poinsot, T., Large-eddy simulation of the shock/turbulence interaction, J. Comput. Phys., 152, 517-549 (1999) · Zbl 0955.76045 |
| [90] | Ducros, F.; Laporte, F.; Soulères, T.; Guinot, V.; Moinat, P.; Caruelle, B., High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows, J. Comput. Phys., 161, 114-139 (2000) · Zbl 0972.76066 |
| [91] | Ducros, F.; Laporte, F.; Soulères, T.; Guinot, V.; Moinat, P.; Caruelle, B., A numerical method for large-eddy simulation in complex geometries, J. Comput. Phys., 197, 215-240 (2004) · Zbl 1059.76033 |
| [92] | Kolgan, V., Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics, J. Comput. Phys., 230, 7, 2384-2390 (2011) · Zbl 1316.76063 |
| [93] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2009), Springer · Zbl 1227.76006 |
| [94] | van Leer, B., A historical oversight: Vladimir P. Kolgan and his high-resolution scheme, J. Comput. Phys., 230, 7, 2378-2383 (2011) · Zbl 1316.65001 |
| [95] | Brugnano, L.; Casulli, V., Iterative solution of piecewise linear systems, SIAM J. Sci. Comput., 30, 463-472 (2007) · Zbl 1155.90457 |
| [96] | Casulli, V.; Zanolli, P., A nested Newton-type algorithm for finite volume methods solving Richards’ equation in mixed form, SIAM J. Sci. Comput., 32, 2255-2273 (2009) · Zbl 1410.76209 |
| [97] | Casulli, V.; Zanolli, P., Iterative solutions of mildly nonlinear systems, J. Comput. Appl. Math., 236, 3937-3947 (2012) · Zbl 1251.65073 |
| [98] | Casulli, V.; Zanolli, P., A nested Newton-type algorithm for finite volume methods solving Richards’ equation in mixed form, SIAM J. Sci. Comput., 32, 4, 2255-2273 (2010) · Zbl 1410.76209 |
| [99] | Bermúdez, A.; Vázquez-Cendón, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 23, 8, 1049-1071 (1994) · Zbl 0816.76052 |
| [100] | Gallardo, J.; Parés, C.; Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys., 227, 574-601 (2007) · Zbl 1126.76036 |
| [101] | Canestrelli, A.; Siviglia, A.; Dumbser, M.; Toro, E., Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed, Adv. Water Resour., 32, 6, 834-844 (2009) |
| [102] | Müller, L. O.; Parés, C.; Toro, E. F., Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties, J. Comput. Phys., 242, 53-85 (2013) · Zbl 1323.92066 |
| [103] | Müller, L. O.; Toro, E. F., Well-balanced high-order solver for blood flow in networks of vessels with variable properties, Int. J. Numer. Methods Biomed. Eng., 29, 12, 1388-1411 (2013) |
| [104] | Käppeli, R.; Mishra, S., Well-balanced schemes for the Euler equations with gravitation, J. Comput. Phys., 259, 199-219 (2014) · Zbl 1349.76345 |
| [105] | Gaburro, E.; Castro, M. J.; Dumbser, M., Well-balanced arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity, Mon. Not. R. Astron. Soc., 477, 2, 2251-2275 (2018) |
| [106] | Abgrall, R., A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes, J. Comput. Phys., 372, 640-666 (2018) · Zbl 1415.76442 |
| [107] | Müller, L. O.; Blanco, P. J., A high order approximation of hyperbolic conservation laws in networks: application to one-dimensional blood flow, J. Comput. Phys., 300, 423-437 (2015) · Zbl 1349.76945 |
| [108] | Barth, T.; Jespersen, D., The design and application of upwind schemes on unstructured meshes (1989), Tech. rep. |
| [109] | Alastruey, J.; Khir, A. W.; Matthys, K. S.; Segers, P.; Sherwin, S. J.; Verdonck, P. R.; Parker, K. H.; Peiró, J., Pulse wave propagation in a model human arterial network: assessment of 1-d visco-elastic simulations against in vitro measurements, J. Biomech., 44, 12, 2250-2258 (2011) |
| [110] | Blanco, P. J.; Watanabe, S. M.; Dari, E. A.; Passos, M. A.; Feijóo, R. A., Blood flow distribution in an anatomically detailed arterial network model: criteria and algorithms, Biomech. Model. Mechanobiol., 13, 1303-1330 (2014) |
| [111] | Busto, S.; Dumbser, M.; Río-Martín, L., An arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations, Appl. Math. Comput., 437, Article 127539 pp. (2023) · Zbl 1510.76081 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
