Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries

• Published in: Journal of computational physics Vol. 386; no. C; pp. 405 - 427
• Main Authors: Petrella, M., Tokareva, S., Toro, E.F.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-06-2019; Elsevier Science Ltd; Elsevier
• Subjects: 60 APPLIED LIFE SCIENCES; ADER Schemes; Arteries; Blood Flow; Burgers equation; Computational physics; Confidence intervals; Flow velocity; Hyperbolic Equations; Hyperbolic systems; Initial conditions; MATHEMATICS AND COMPUTING; Methodology; Parameter uncertainty; Pressure effects; Solvers; Stochastic Finite Volume Method; Uncertainty quantification; Uncertainty Quantification; Variation; Veins & arteries; Uncertainty quantification; Hyperbolic equations; ADER schemes; Stochastic finite volume method; Blood flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.02.013

We present a Stochastic Finite Volume - ADER (SFV-ADER) methodology for Uncertainty Quantification (UQ) in the general framework of systems of hyperbolic balance laws with uncertainty in parameters. The resulting method is weakly intrusive, meaning that deterministic solvers need minor modifications to include random contributions, and has no theoretical accuracy barrier. An illustration of the second-order version for the viscous Burgers equation with uncertain viscosity coefficient is first given in detail, under different deterministic initial conditions; the attainment of the theoretically expected convergence rate is demonstrated empirically; results and features of our scheme are discussed. We then extend the SFV-ADER method to a non-linear hyperbolic system with source terms that models one-dimensional blood flow in arteries, assuming uncertainties in a set of parameters of the problem. Results are then compared with measurements in a well-defined 1:1 experimental replica of the network of largest arteries in the human systemic circulation. For a 99%-confidence level, severe influence of parameter fluctuations is seen in pressure profiles, while secondary effects on flow rate profiles become visible, particularly in terminal branches. Results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws.
•No theoretical accuracy barrier.•Uncertain boundary conditions included.•Source terms included.