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SUMMARYThe two-dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two-dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second-order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case.The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant-Friedrichs-Lewy (CFL) condition.

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SUMMARYFriction is one of the relevant forces included in the momentum equation of the 1D shallow-water model. This work shows that a pointwise discretization of the friction term unbalances this term with the rest of the terms in the equation in steady state. On the other hand, an upwind discretization of the friction term ensures the correct discrete balance. Furthermore, a conservative technique based on the limitation of the friction value is proposed in order to avoid unbounded values of the friction term in unsteady cases of advancing front over dry and rough surfaces. This limitation improves the quality of unsteady solutions in wet/dry fronts and guarantees the numerical stability in cases with dominant friction terms. The proposed discretization is validated in some test cases with analytical solution or with measured data and used in some practical cases.

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Mathematical models are widely recognized as a valuable tool for cardiovascular diagnosis and the study of circulatory diseases, especially to obtain data that require otherwise invasive measurements. To correctly simulate body hemodynamics, the viscoelastic properties of vessels walls are a key aspect to be taken into account as they play an essential role in cardiovascular behavior. The present work aims to apply the augmented fluid-structure interaction system of blood flow to real case studies to assess the validity of the model as a valuable resource to improve cardiovascular diagnostics and the treatment of pathologies. First, the ability of the model to correctly simulate pulse waveforms in single arterial segments is verified using literature benchmark test cases. Such cases are designed taking into account a simple elastic behavior of the wall in the upper thoracic aorta and in the common carotid artery. Furthermore, in-vivo pressure waveforms, extracted from tonometric measurements performed on four human common carotid arteries and two common femoral arteries, are compared to numerical solutions. It is highlighted that the viscoelastic damping effect of arterial walls is required to avoid an overestimation of pressure peaks. An effective procedure to estimate the viscoelastic parameters of the model is herein proposed, which returns hysteresis curves of the common carotid arteries dissipating energy fractions in line with values calculated from literature hysteresis loops in the same vessel.variations are determined by the physical and mechanical properties of blood and vessels walls, which are the essence of a complex fluid-structure interaction (FSI) mechanism, as well as by the anatomy of the entire cardiac network [33,48]. Viscoelastic properties of vessels play an essential role in the cardiovascular behavior [39,33,20]. In fact, viscoelasticity is one of the features that must be realistically included in the mathematical model when accurate numerical results are sought [20,1,26]. Vessel walls manifest viscoelastic properties that are summed up in three main attributes: creep, stress relaxation and hysteresis [6,39,22]. Among the existing linear viscoelastic models, the Standard Linear Solid (SLS) model provides a better representation of the arterial wall mechanics than the generally adopted Kelvin-Voigt model [1,26,46,32,23], being the latter unable to describe an exponential decay of stress over time [49,10,45]. On the other hand, when modeling the vessel mechanics by means of an elastic behaviour, the information related to hysteresis (i.e. the energy dissipated by viscoelastic effects) vanishes and pressure peaks are overestimated [1,6,20].The augmented FSI (a-FSI) system for blood flow modeling, presented in [7,8], is herein extended to real case studies in single arteries, to assess the capability of the model to serve as a valuable tool for practical medical applications, cardiovascular diagnosis and the study of circulatory pathologies. The extension of the model underlines the importanc...