In this article, we show how to construct a numerical method for solving hyperbolic problems, whether linear or nonlinear, using a continuous representation of the variables and their mean value in each triangular element. This type of approach has already been introduced by Roe, and others, in the multidimensional framework under the name of Active flux, see \cite{AF1,AF2,AF3,AF4,AF5}. Here, the presentation is more general and follows \cite{Abgrall_AF,BarzukowAbgrall}. { Various} examples show the good behavior of the method in both linear and nonlinear cases, including non-convex problems. The expected order of precision is obtained in both the linear and nonlinear cases. This work represents a step towards the development of methods in the spirit of virtual finite elements for linear or nonlinear hyperbolic problems, including the case where the solution is not regular.

翻译：本文展示了如何利用变量在三角形单元内的连续表示及其平均值，构建求解双曲问题（无论线性或非线性）的数值方法。此类方法已由Roe等人在多维框架下以"主动通量"（Active flux）为名提出（见参考文献\cite{AF1,AF2,AF3,AF4,AF5}）。本文的表述更为通用，并遵循文献\cite{Abgrall_AF,BarzukowAbgrall}的框架。多种算例表明，该方法在线性与非线性情形（包括非凸问题）中均具有良好的表现。在线性与非线性情况下均能达到预期的精度阶数。这项工作为发展面向线性/非线性双曲问题（包含非正则解情形）的虚拟有限元类方法迈出了重要一步。