Several numerical methods have been developed to analyze water hammer dynamics, and among these, the method of characteristics (MOC) is the most widely applied, establishing the time-dependent characteristic equations that account for the components of the unsteady flow. This paper introduces a matrix formulation of the characteristic equations, which is based on an assumed initial incidence matrix that describes the topology of the network, including loops. Unlike traditional MOC, the matrix equations are used to simultaneously solve for all pressures and flows in the network, at each time step, as a linear function of the pressure and flow at all locations in the network in the previous time step. Two solution procedures are proposed, both of which solve for pressures by decomposing the linear systems of equations into a reduced linear system of equations that is on the order of the number of nodes, and use the resulting pressures to update the flow vector, that is on the order of the number of pipes or pipe reaches, at each time step. The proposed solution procedures are not dependent to the network topologies. For different shapes of pipe network, the formulation remains unchanged and the user need only enters different input data in the form of vectors to find the solution using matrix–vector multiplications at each time step. Fast linear solvers can also be implemented to speed up the process, because the linear system of equations at the core of these algorithms is a Stieltjes matrix. These solution procedures are applied for matrix formulations of two numerical examples, and the resulting nodal pressure and pipe flow at each time step are nearly identical.

中文翻译：

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管网水锤分析的统一矩阵框架

已经开发了几种数值方法来分析水锤动力学，其中，特征法 (MOC) 是应用最广泛的方法，它建立了解释非定常流动分量的瞬态特征方程。本文介绍了特征方程的矩阵公式，该公式基于一个假设的初始关联矩阵，该矩阵描述了网络的拓扑结构，包括环路。与传统的 MOC 不同，矩阵方程用于在每个时间步同时求解网络中的所有压力和流量，作为前一时间步中网络中所有位置的压力和流量的线性函数。提出了两种解决方法，两者都通过将线性方程组分解为节点数数量级的简化线性方程组来求解压力，并使用产生的压力更新流向量，即数量级管道或管道到达，在每个时间步长。建议的解决方案过程不依赖于网络拓扑。对于不同形状的管网，公式保持不变，用户只需要以向量的形式输入不同的输入数据，就可以在每个时间步使用矩阵-向量相乘来找到解决方案。还可以实施快速线性求解器来加速该过程，因为这些算法核心的线性方程组是 Stieltjes 矩阵。这些求解过程适用于两个数值例子的矩阵公式，