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Analysis of a geometrical multiscale blood flow model based on the coupling of ODE's and hyperbolic PDE's

Miguel Angel Fernández 1 Vuk Milisic 2 Alfio Quarteroni 3
1 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6
Abstract : For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions are becoming common practice [L. Formaggia et al., Comput. Vis. Sci., 2 (1999), pp. 75-83, A. Quarteroni and A. Veneziani, Multiscale Model. Simul., 1 (2003), pp. 173-195, L. Formaggia et al., Comput. Methods Appl. Mech. Engrg., 191 (2001), pp. 561-582]. In this paper we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional (0D) model, describing the global characteristics of the circulatory system, and a one-dimensional (1D) model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODEs and the 1D hyperbolic system of PDEs) ; then we use fixed-point techniques. The abstract result is then applied to the original blood flow case under very realistic hypotheses on the data. This work represents the 1D-0D counterpart of the 3D-0D mathematical analysis reported in [A. Quarteroni and A. Veneziani, Multiscale Model. Simul., 1 (2003), pp. 173-195].
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https://hal.inria.fr/hal-00705109
Contributor : Miguel Angel Fernández <>
Submitted on : Wednesday, June 6, 2012 - 7:07:46 PM
Last modification on : Saturday, March 28, 2020 - 2:23:52 AM

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Miguel Angel Fernández, Vuk Milisic, Alfio Quarteroni. Analysis of a geometrical multiscale blood flow model based on the coupling of ODE's and hyperbolic PDE's. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2005, 4 (1), pp.215-236. ⟨10.1137/030602010⟩. ⟨hal-00705109⟩

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