We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending Finite Volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of an electrical treeing.

翻译：本文提出了在一维图边定义问题的近似数值求解方案。我们重点研究线性输运方程和漂移-扩散方程，通过将迎风通量的有限体积格式扩展至具有任意数量进出边的分岔节点域，并采用隐式时间离散方法进行方程离散。我们证明了离散问题存在唯一正解，并在电气树枝状结构的复杂几何模型上对所提方法进行了验证。