标题： 虚拟有限元和双曲问题：PAMPA算法 显示英文标题

标题： Virtual finite element and hyperbolic problems: the PAMPA algorithm

摘要： 在本文中，我们探讨了使用虚拟单元方法的概念来在一般的多边形网格上求解标量和系统双曲问题。 新的方案源自活跃通量方法\cite{AF1}，该方法结合了在单元边界处的点值的使用与一个额外的自由度，该自由度表示每个控制体积内的解的平均值。 沿着在\cite{Abgrall_AF,abgrall2023activefluxtriangularmeshes} \red{that integrate the active flux technique} 中引入的残差分布方案族的思路，我们设计了新颖的三阶精度方法，这些方法依靠VEM技术通过多项式无关的近似来离散数值解的梯度，\red{by}采用为每个单元局部定义的虚拟基。 得到的离散化是全局连续的，对于非线性问题需要由\new{a monolithic convex limiting strategy extended from \cite{Abgrall_BP_PAMPA}}提供的稳定化处理。 这适用于离散解的点值和平均值。 我们展示了对标量问题以及二维声学和欧拉方程的应用。 我们通过一系列涉及光滑解、激波和其他不连续性的基准测试来评估所提出方案的准确性和鲁棒性。 显示英文摘要

摘要： 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} \red{that integrate 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, \red{by} 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 \new{a monolithic convex limiting strategy extended from \cite{Abgrall_BP_PAMPA}}. 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 two dimension. The accuracy and the robustness of the proposed schemes are assessed against a suite of benchmarks involving smooth solutions, shock waves and other discontinuities.