Abstract A new formulation for isothermal gas flow in a pipeline with gravity and friction source terms is introduced and solved by finite element methods using FEniCS open-source computing platform. The proposed formulation can be easily adapted to simulate different types of real boundary conditions and involves only one scalar unknown instead of two ones (typically mass flux and pressure or density). The main motivation for presenting the work is to find a compact formulation which is able to be solved with a high order accuracy scheme just by changing the type of elements. Also, it can be extended to gas network easily. In order to check the performance of the numerical method, it has been applied to several test problems. At first, with the purpose of finding the order of accuracy, two manufactured tests for different types of boundary conditions are solved. The results show that, for nodal finite elements of degree one, the order of accuracy in space is more than two and for time is between two and three. Then, two test cases with step boundary conditions are solved. The corresponding numerical results show that, even with large time-step and coarse mesh, they are in good agreement with those obtained with a fine mesh. Finally, a real pipe from the Spanish gas network is solved and the results compared with on-line measurements.

中文翻译：

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含源项的管内气体流动新公式的有限元解

摘要 引入了具有重力和摩擦源项的管道中等温气流的新公式，并使用FEniCS开源计算平台通过有限元方法求解。建议的公式可以很容易地适应模拟不同类型的真实边界条件，并且只涉及一个未知标量而不是两个标量（通常是质量通量和压力或密度）。展示这项工作的主要动机是找到一个紧凑的公式，只需改变元素的类型，就可以用高阶精度方案来解决。此外，它可以轻松扩展到天然气网络。为了检验数值方法的性能，它已被应用于几个测试问题。起初，为了找到准确度的顺序，解决了针对不同类型边界条件的两个制造测试。结果表明，对于一阶节点有限元，空间精度在二阶以上，时间精度在二到三阶之间。然后，求解两个具有阶跃边界条件的测试用例。相应的数值结果表明，即使是大的时间步长和粗网格，它们与细网格得到的结果也有很好的一致性。最后，解决了来自西班牙天然气管网的真实管道，并将结果与​​在线测量结果进行了比较。即使是大的时间步长和粗网格，它们也与用细网格获得的结果非常吻合。最后，解决了来自西班牙天然气管网的真实管道，并将结果与​​在线测量结果进行了比较。即使是大的时间步长和粗网格，它们也与用细网格获得的结果非常吻合。最后，解决了来自西班牙天然气管网的真实管道，并将结果与​​在线测量结果进行了比较。