Well-balanced discontinuous Galerkin methods for the one-dimensional blood flow through arteries model with man-at-eternal-rest and living-man equilibria

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

“…(ii) It is well-balanced, that is, it preserves the discrete version of the man-at-eternal-rest (5).…”

“…We choose several numerical test problems mainly from References [5,10]. The purpose is to exhibit the performance of the proposed scheme, in particular for testing the well-balanced property.…”

“…The purpose of this test problem is to check that the proposed scheme preserves the zero pressure man-at-eternal-rest steady state (5) with C = 0. The configuration exhibits no flow and includes a change in the section of the artery.…”

“…This example (see [5]) corresponds to the case of a dead man with stenosis. The cross-sectional area at rest is A 0 (x) = 𝜋R 2 0 (x), where…”

“…This example (see [5]) corresponds to the case of a dead man with stenosis. The cross‐sectional area at rest is $$$ {A}_0(x)=\pi {R}_0^2(x) $$$, where $$$$ {R}_0(x)=\left\{\begin{array}{cc}\tilde{R}+\Delta R& \mathrm{for}\ x\in \left[0,{x}_1\right]\cup \left[{x}_4,L\right],\\ {}\tilde{R}-\frac{\Delta R}{2}\left(\sin \left(\frac{x-{x}_1}{x_2-{x}_1}\pi -\frac{\pi }{2}\right)-1\right]& \mathrm{for}\ x\in \left({x}_1,{x}_2\right),\\ {}\tilde{R}& \mathrm{for}\ x\in \left[{x}_2,{x}_3\right],\\ {}\tilde{R}-\frac{\Delta R}{2}\left(\cos \left(\frac{x-{x}_3}{x_4-{x}_3}\pi \right)-1\right)& \mathrm{for}\ x\in \left({x}_3,{x}_4\right)\end{array}\right.$$…”