Virtual finite element and hyperbolic problems: the PAMPA algorithm

TL;DR

This paper introduces a novel third-order accurate Virtual Element Method-based scheme for solving hyperbolic problems on polygonal grids, combining active flux concepts with stabilization techniques for improved robustness and accuracy.

Contribution

It develops a new VEM-based active flux scheme with polynomial-free gradient approximation and stabilization, extending the active flux approach to hyperbolic PDEs on polygonal meshes.

Findings

Achieves third-order accuracy on benchmark problems.

Demonstrates robustness with shock and discontinuity handling.

Validates effectiveness on acoustics and Euler equations.

Abstract

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.