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

. A well-known reduced model of the ﬂow of blood in arteries can be formulated as a strictly hyperbolic system of two scalar balance law 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, The numerical viscosity of entropy stable schemes for systems of conservation laws, I , Math. Comput., 49 (1987) pp. 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 computationally inexpensive and easy-to-implement explicit entropy stable scheme for the blood low model. It is proven that the scheme is well-balanced (i.e., preserves certain steady solutions of the model) and numerical examples are presented. = P ( A ) is the internal pressure and F ( x, t ) is the friction force per unit length. The present work is restricted to the inviscid limit where F = 0 [2]. The system (1.1) is closed by a relationship of the form (see [3]) P = P ext + β ( A 1 / 2 − A 1 / 2 0 ), where P ext stands for the external pressure (assumed to be constant), the coeﬃcient β , which is supposedly constant as well, accounts for the material properties of the elastic vessel, and A 0 = A 0 ( x ) is the vessel