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.

中文翻译：

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具有永恒休息和活人平衡的一维血流通过动脉模型的平衡不连续伽辽金方法

摘要 动脉中的血流运动可以通过守恒定律系统建模，并且在医学环境中具有广泛的应用。在本文中，我们为一维血流模型提出了有效的、均衡的不连续 Galerkin 方法，该方法保留了永恒静止的人（零速度）和更一般的活人（非零速度）平衡。良好平衡状态的恢复、将数值解分解为平衡和波动部分以及适当的源项和数值通量近似是设计良好平衡方法的关键思想。给出了数值例子来验证良好的平衡特性、高阶精度、平滑和不连续解的良好分辨率，以及很好地捕获近平衡解的能力。我们还在具有各种夏皮罗数的近平衡流上测试了所提出的方法。Man-at-eternal-rest well-balanced 方法适用于低 Shapiro 数的问题，但当 Shapiro 数变大时会产生虚假流，而活人良好平衡方法对所有范围的 Shapiro 数都表现良好。