Finite volume and WENO scheme in one-dimensional vascular system modelling

Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow

“…The results show a forward moving pressure wave, with positive flow rate, which reaches the end of the domain and gets reflected. The reflected wave is characterized by negative values of the pressure and positive flow rates [12,25]. The results obtained with the kinematically coupled scheme are in excellent agreement with those obtained in [25] using an implicit scheme.…”

A ‘well‐balanced’ finite volume scheme for blood flow simulation

“…We will prefer a wellbalanced method inspired from the hydrostatic reconstruction ( [47,48]). As for the hydrostatic reconstruction, we do not consider the friction term, and we are looking for a scheme, which preserves steady states at rest (16), the second equation in (16) means that we have…”

“…At first, we derive the equations written in conservative form; looking in the literature, we found that it has been only written by Wibmer in a PhD thesis and by Cavallini et al . , but not exploited in the ‘well‐balanced’ point of view. Mostly , equations are in fact written in a non‐conservative form.…”

“…Since then, the pulsatile wave flow has been understood and is explained in classical books such as Lighthill [3] and Pedley [4]. More recently, much work has been done and many methods of resolution have been proposed to handle this set of equations, among them [1,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], and we forgot many actors. Even if recent progress of fluid structure interaction in 3D have been performed ( [21,22], among others), those advances have not made the 1D modelization obsolete.…”

Well‐balanced finite difference weighted essentially non‐oscillatory schemes for the blood flow model

“…In recent years, there have been many interesting attempts proposed in the literature to derive well-balanced schemes for the blood flow model. For example, Delestre et al [25] present a wellbalanced finite volume scheme for the blood flow model based on the conservative governing equations [26][27][28]. Recently, Müller et al [29] constructed a well-balanced high order finite volume for the blood flow in elastic vessels with varying mechanical properties.…”