One-Dimensional Blood Flow with Discontinuous Properties and Transport: Mathematical Analysis and Numerical Schemes (Communications in Computational Physics, 2021)
In this paper we consider the one-dimensional blood flow model with discontinuous mechanical and geometrical properties, as well as passive scalar transport,proposed in [E.F. Toro and A. Siviglia. Flow in collapsible tubes with discontinuousmechanical properties: mathematical model and exact solutions. Communications inComputational Physics. 13(2), 361-385, 2013], completing the mathematical analysis byproviding new propositions and new proofs of relations valid across different waves.Next we consider a first order DOT Riemann solver, proposing an integration path thatincorporates the passive scalar and proving the well-balanced properties of the resulting numerical scheme for stationary solutions. Finally we describe a novel and simplewell-balanced, second order, non-linear numerical scheme to solve the equations under study; by using suitable test problems for which exact solutions are available, weassess the well-balanced properties of the scheme, its capacity to provide accurate solutions in challenging flow conditions and its accuracy.
Keywords: Blood flows; Riemann problem; wave relations; finite volume method; well-balancing.