Numerical approximation and uncertainty quantification for arterial blood flow models with viscoelasticity

Abstract

The importance of the study of the blood flow equations is widely recognized as it is a tool to understand the circulatory system. Arteries and veins result to have both elastic and viscous behaviour. Models for the first case are much more studied as they result to be simpler and still satisfying if compared to experimental data. In this paper, we consider a model which encompasses both the elastic and viscoelastic response in arterial walls, respectively leading to a conservative and a non-conservative system. We present a second-order scheme based on the first-order Price-T scheme and the MUSCL-Hancock strategy. This approach automatically adapts to the above conservative and non-conservative cases.Then, we perform a Sensitivity Analysis (SA) based on the Continuous Sensitivity Equation Method (CSEM), whose aim is the study of how changes in the inputs of a model can affect its outputs. In particular, the sensitivity is defined as the derivative (with respect to an uncertain parameter a) of the solution of the system taken into consideration. Since the CSEM cannot be directly applied to discontinuous solutions, we add a source term to compensate the spikes associated to the Dirac delta functions that can arise in the sensitivity variables.One of the main applications of SA is uncertainty quantification, which is investigated for a Riemann problem as well as for a network of 37 arteries. Details on junctions for coupling two or more vessels are also given.