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Some new properties of the PamPa scheme

Rémi Abgrall, Philipp Öffner, Yongle Liu

TL;DR

Active Flux/PamPa schemes for hyperbolic conservation laws are reinterpreted as DG steps with a subsequent projection to a continuous space, enabling a DG-like high-order framework. The work proves intrinsic bound-preserving properties for several PamPa variants and demonstrates a one-dimensional SBP structure, linking PamPa to energy-stable SBP theory. It also extends the construction to non-constant advection and nonlinear problems, and discusses VEM-based polygonal-mesh extensions, providing a robust theoretical foundation for invariant-domain preserving high-order schemes on general meshes. Collectively, these results offer a principled approach to designing stable, high-order PamPa schemes with provable positivity and SBP characteristics, and guide boundary-condition discretisations in complex geometries.

Abstract

In this paper, we provide a few new properties of Active Flux (AF)/Point-Average-Moment PolynomiAl-interpreted (\pampa) schemes. First, we show, in full generality, that the AF/pampa schemes can be interpreted in such a way that the discontinuous Galerkin (dG) scheme is one of their building blocks. Secondly we provide intrinsic bound preserving properties of the current variant of pampa. This is also illustrated numerically. Last, we show, at least in one dimension, that the pampa scheme has the summation by part (SBP) property.

Some new properties of the PamPa scheme

TL;DR

Active Flux/PamPa schemes for hyperbolic conservation laws are reinterpreted as DG steps with a subsequent projection to a continuous space, enabling a DG-like high-order framework. The work proves intrinsic bound-preserving properties for several PamPa variants and demonstrates a one-dimensional SBP structure, linking PamPa to energy-stable SBP theory. It also extends the construction to non-constant advection and nonlinear problems, and discusses VEM-based polygonal-mesh extensions, providing a robust theoretical foundation for invariant-domain preserving high-order schemes on general meshes. Collectively, these results offer a principled approach to designing stable, high-order PamPa schemes with provable positivity and SBP characteristics, and guide boundary-condition discretisations in complex geometries.

Abstract

In this paper, we provide a few new properties of Active Flux (AF)/Point-Average-Moment PolynomiAl-interpreted (\pampa) schemes. First, we show, in full generality, that the AF/pampa schemes can be interpreted in such a way that the discontinuous Galerkin (dG) scheme is one of their building blocks. Secondly we provide intrinsic bound preserving properties of the current variant of pampa. This is also illustrated numerically. Last, we show, at least in one dimension, that the pampa scheme has the summation by part (SBP) property.

Paper Structure

This paper contains 19 sections, 5 theorems, 173 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction.
  2. Reinterpretation of the PamPa scheme.
  3. Standard PamPa Scheme.
  4. High order PamPa scheme.
  5. PamPa for triangles.
  6. Application to the convection problem with constant advection speed.
  7. Interpretation of the PamPa scheme.
  8. Generatisation to non constant advection speeds and non linear problems.
  9. Intrinsic positivity properties of PamPa
  10. The 1D case.
  11. The 2D case.
  12. Polynomial case: quadratic and cubic cases of Abgrall2023b.
  13. Polynomial case: cubic moment.
  14. Approximation using Virtual finite element (VEM) approximation
  15. Analysis
  16. ...and 4 more sections

Key Result

Lemma 2.1

The linear forms linear form points-linear forms moment are uni-solvant on $V$.

Figures (4)

  • Figure 1: Interpolation points for the triangular and quadrangular, cubic case.
  • Figure 2: Solution of $u_t+u_x=0$ with periodic boundary conditions on $[0,1]$ for the initial condition $u_0=\cos(2\pi x)$ after $10$ and $100$ periods, compared to the exact solution, on regular mesh (fig. a). On fig. (b), the initial solution is $u_0=e^{-10x^2}$ on $[-1,1]$ after 10 rotations on a random mesh.
  • Figure 3: Jiang and Shu's problem. (a): average values, (b): point values. Original: scheme of BP_Pampa_VEM, Numerical: the BP strategy is applied only on the point values.
  • Figure 4: Solution of Lax-Liu case # 3.

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Dual basis
  • proof
  • Remark 2.3
  • Remark 2.4: Scheme for quadrangles
  • Remark 2.5: Scheme for Polygons
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • ...and 3 more