In this paper we propose a method for coupling distributed and lumped models for the blood circulation. Lumped parameter models, based on an analogy between the circulatory system and an electric or a hydraulic network are widely employed in the literature to investigate different systemic responses in physiologic and pathologic situations (see e.g. [13, 24, 30, 15, 4, 27, 11, 14]). From the mathematical viewpoint these models are represented by ordinary differential equations. On the other hand, for the accurate description of local phenomena, the Navier–Stokes equations for incompressible fluids are considered. In the multiscale perspective, lumped models have been adopted (see e.g. [16]) as a numerical preprocessor to provide a quantitative estimate of the boundary conditions at the interfaces. However, the two solvers (i.e. the lumped and the distributed one) have been used separately. In the present work, we introduce a genuinely heterogeneous multiscale approach where the local model and the systemic one are coupled at a mathematical and numerical level and solved together. In this perspective, we have no longer boundary conditions on the artificial sections, but interface conditions matching the two submodels. The mathematical model and its numerical approximation are carefully addressed and several test cases are considered
a b s t r a c tWe consider splitting methods for the numerical integration of separable non-autonomous differential equations. In recent years, splitting methods have been extensively used as geometric numerical integrators showing excellent performances (both qualitatively and quantitatively) when applied on many problems. They are designed for autonomous separable systems, and a substantial number of methods tailored for different structures of the equations have recently appeared. Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly. We show that both procedures correspond to introducing the time as two new coordinates. We generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques. We show that the performance as well as the order of the final method can strongly depend on the particular choice. We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems. This technique is applied to different problems and its performance is illustrated for several numerical examples.
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