A well‐balanced and entropy stable scheme for a reduced blood flow model

Abstract

A well‐known reduced model of the flow of blood in arteries can be formulated as a strictly hyperbolic system of two scalar balance laws in one space dimension where the unknowns are the cross‐sectional area of the artery and the average blood flow velocity as functions of the axial coordinate and time. This system is endowed with an entropy pair such that solutions of the balance equations satisfy an entropy inequality in the distributional sense. It is demonstrated that this property can be utilized to construct an entropy stable finite difference scheme for the blood flow model based on the general framework by Tadmor (E. Tadmor, Math. Comput., 49 (1987), 91–103). Furthermore, a fourth‐order extension of the resulting entropy conservative flux and a fourth‐order sign‐preserving reconstruction of the scaled entropy variables are employed as well as a second‐order strong stability preserving Runge–Kutta method for time discretization. The result is a computationally inexpensive and easy‐to‐implement explicit entropy stable scheme for the blood flow model. It is proven that the scheme is well‐balanced (i.e., preserves certain steady solutions of the model) and numerical examples are presented.

A Steady-State-Preserving Numerical Scheme for One-Dimensional Blood Flow Model

“…We now turn to the right-hand side. Using (10), the numerical flux F j+1/2 explicitly given by (17), Equation (18), and the definition of G j+1/2 (18), we obtain…”