In this paper, we explore the use of the Virtual Element Method concepts to solve scalar and system hyperbolic problems on general polygonal grids. The new schemes stem from the active flux approach \cite{AF1}, which combines the usage of point values at the element boundaries with an additional degree of freedom representing the average of the solution within each control volume. Along the lines of the family of residual distribution schemes introduced in \cite{Abgrall_AF,abgrall2023activefluxtriangularmeshes} to bridge the active flux technique, we devise novel third order accurate methods that rely on the VEM technology to discretize gradients of the numerical solution by means of a polynomial-free approximation, hence adopting a virtual basis that is locally defined for each element. The obtained discretization is globally continuous, and for nonlinear problems it needs a stabilization which is provided by the \textit{a posteriori} MOOD paradigm \cite{Mood1}. This is applied to both point and average values of the discrete solution. We show applications to scalar problems, as well as to the acoustics and Euler equations in 2D. The accuracy and the robustness of the proposed schemes are assessed against a suite of benchmarks involving smooth solutions, shock waves and other discontinuities.

翻译：本文探讨了利用虚拟元方法概念在一般多边形网格上求解标量与系统双曲问题的应用。新方案源于主动通量方法\cite{AF1}，该方法结合了单元边界处的点值以及代表每个控制体积内解的平均值的附加自由度。遵循\cite{Abgrall_AF,abgrall2023activefluxtriangularmeshes}中引入的残差分布格式族以桥接主动通量技术的思路，我们设计了新颖的三阶精度方法，这些方法依赖VEM技术通过无多项式逼近来离散数值解的梯度，从而采用为每个单元局部定义的虚拟基。所获得的离散化是全局连续的，对于非线性问题，需要由\textit{后验}MOOD范式\cite{Mood1}提供的稳定化处理。这被应用于离散解的点值和平均值。我们展示了该方法在标量问题以及二维声学和欧拉方程中的应用。通过一系列涉及光滑解、激波和其他间断的基准测试，评估了所提格式的精度和鲁棒性。