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Virtual finite element and hyperbolic problems: the PAMPA algorithm

Rémi Abgrall, Yongle Liu, Walter Boscheri

TL;DR

The paper presents a high-order, globally continuous method for hyperbolic PDEs on general polygonal meshes by fusing Virtual Element Method concepts with the Active Flux approach (PAMPA). It replaces explicit polynomial bases with a polynomial-free gradient discretization and stabilizes nonlinear problems via a monolithic convex limiting strategy, extended from 1D/triangular precedents and augmented by MOOD-like ideas. The schemes are demonstrated on scalar convection, KPP-type problems, acoustics, and Euler equations, with robust performance on triangles, quadrilaterals, and general polygons, including moving vortices and complex shock interactions. The framework offers flexibility in mesh design and promises extensions to higher order and adaptive mesh refinement, balancing accuracy and robustness on irregular polygonal grids.

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.

Virtual finite element and hyperbolic problems: the PAMPA algorithm

TL;DR

The paper presents a high-order, globally continuous method for hyperbolic PDEs on general polygonal meshes by fusing Virtual Element Method concepts with the Active Flux approach (PAMPA). It replaces explicit polynomial bases with a polynomial-free gradient discretization and stabilizes nonlinear problems via a monolithic convex limiting strategy, extended from 1D/triangular precedents and augmented by MOOD-like ideas. The schemes are demonstrated on scalar convection, KPP-type problems, acoustics, and Euler equations, with robust performance on triangles, quadrilaterals, and general polygons, including moving vortices and complex shock interactions. The framework offers flexibility in mesh design and promises extensions to higher order and adaptive mesh refinement, balancing accuracy and robustness on irregular polygonal grids.

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.

Paper Structure

This paper contains 30 sections, 129 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Meshes and Approximation space
  3. Meshes
  4. Approximation space
  5. Numerical schemes
  6. High order schemes
  7. Low order schemes
  8. Limiting strategies
  9. A-posteriori limiting strategy
  10. A monolithic convex limiting method
  11. Evolving cell averages.
  12. Evolving point values.
  13. Extension to Euler equations.
  14. Boundary conditions
  15. Average values
  16. ...and 15 more sections

Figures (10)

  • Figure 1: Definitions of $e^\pm$ and ${\mathbf {n}}^\pm$.
  • Figure 2: Sub-triangulation and normals that used for the low order scheme.
  • Figure 3: Notation for the boundary conditions.
  • Figure 4: KPP problem using quadratic approximation. Average values. The CFL is $0.3$.
  • Figure 5: Acoustics: plots of the velocity and the computational mesh.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Remark 3.1
  • Remark 3.2