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

Abstract

The movement of blood flow in arteries can be modeled by a system of conservation laws and has a range of applications in medical contexts. In this paper, we present efficient well-balanced discontinuous Galerkin methods for the one-dimensional blood flow model, which preserve the man-at-eternal-rest (zero velocity) and more general living-man (non-zero velocity) equilibria. Recovery of well-balanced states, decomposition of the numerical solutions into the equilibrium and fluctuation parts, and appropriate source term and numerical flux approximations are the key ideas in designing well-balanced methods. Numerical examples are presented to verify the well-balanced property, high order accuracy, good resolution for both smooth and discontinuous solutions, and the ability to capture nearly equilibrium solutions well. We also test the proposed methods on nearly equilibrium flows with various Shapiro numbers. Man-at-eternal-rest well-balanced methods work well for problems with low Shapiro number, but generate spurious flows when Shapiro number gets larger, while the living-man well-balanced methods perform well for all ranges of Shapiro number.