In this paper, we propose an arbitrarily high-order accurate fully well-balanced numerical method for the one-dimensional blood flow model. The developed method is based on a continuous representation of the solution and a natural combination of the conservative and primitive formulations of the studied PDEs. The degrees of freedom are defined as point values at cell interfaces and moments of the conservative variables inside the cell, drawing inspiration from the discontinuous Galerkin method. The well-balanced property, in the sense of an exact preservation of both the zero and non-zero velocity equilibria, is achieved by a well-balanced approximation of the source term in the conservative formulation and a well-balanced residual computation in the primitive formulation. To lowest (3rd) order this method reduces to the method developed in [Abgrall and Liu, A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations: A Combination of Conservative and Primitive Formulations, arXiv preprint, arXiv:2304.07809]. Several numerical tests are shown to prove its well-balanced and high-order accuracy properties.

翻译：本文提出了一种针对一维血流模型的任意高阶精确全平衡数值方法。该方法基于解的连续表示，并自然结合了所研究偏微分方程的守恒形式与原始形式。自由度定义为单元界面处的点值及单元内守恒变量的矩，其灵感来源于间断伽辽金方法。平衡性质（即精确保持零速度与非零速度两种均衡态）通过守恒形式中源项的平衡近似以及原始形式中残差的平衡计算实现。当精度降至三阶时，该方法退化为 [Abgrall 和 Liu 在《浅水方程平衡格式设计新方法：守恒形式与原始形式的结合》（arXiv 预印本，arXiv:2304.07809）] 中提出的方法。多项数值测试验证了该方法的平衡性与高阶精度特性。