Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust PAMPA Scheme in the DG Formulation on Unstructured Triangular Meshes: bound preservation, oscillation elimination, and boundary conditions

Rémi Abgrall, Yongle Liu

TL;DR

The paper addresses high-order numerical schemes for hyperbolic conservation laws on unstructured triangular meshes that simultaneously achieve bound preservation and non-oscillatory behavior. It develops a PAMPA scheme embedded in a DG framework by constructing a projection that recovers PAMPA from DG and applying a monolithic convex limiting to enforce invariants. A novel convex oscillation-eliminating parameter based on high-order derivative jumps is extended to multidimensional problems and combined with the bound-preserving parameter via a minimum, yielding a robust BP OE blend. Truncation-error analysis certifies third-order accuracy for smooth solutions ($O(h^3)$), and extensive tests on scalar and Euler equations confirm bound preservation, non-oscillatory performance, and practical robustness on unstructured meshes.

Abstract

We propose an improved version of the PAMPA algorithm where the solution is sought as globally continuous. The scheme is locally conservative, and there is no mass matrix to invert. This method had been developed in a series of papers, see e.g \cite{Abgrall2024a} and the references therein. In \cite{Abgrall2025d}, we had shown the connection between PAMPA and the discontinuous Galerkin method, for the linear hyperbolic problem. Taking advantage of this reinterpretation, we use it to define a family of methods, show how to implement the boundary conditions in a rigorous manner. In addition, we propose a method that complements the bound preserving method developed in \cite{Abgrall2025d} in the sense that it is non oscillatory. A truncation error analysis is provided, it shows that the scheme should be third order accurate for smooth solutions. This is confirmed by numerical experiments. Several numerical examples are presented to show that the scheme is indeed bound preserving and non oscillatory on a wide range on numerical benchmarks.

Robust PAMPA Scheme in the DG Formulation on Unstructured Triangular Meshes: bound preservation, oscillation elimination, and boundary conditions

TL;DR

The paper addresses high-order numerical schemes for hyperbolic conservation laws on unstructured triangular meshes that simultaneously achieve bound preservation and non-oscillatory behavior. It develops a PAMPA scheme embedded in a DG framework by constructing a projection that recovers PAMPA from DG and applying a monolithic convex limiting to enforce invariants. A novel convex oscillation-eliminating parameter based on high-order derivative jumps is extended to multidimensional problems and combined with the bound-preserving parameter via a minimum, yielding a robust BP OE blend. Truncation-error analysis certifies third-order accuracy for smooth solutions (), and extensive tests on scalar and Euler equations confirm bound preservation, non-oscillatory performance, and practical robustness on unstructured meshes.

Abstract

We propose an improved version of the PAMPA algorithm where the solution is sought as globally continuous. The scheme is locally conservative, and there is no mass matrix to invert. This method had been developed in a series of papers, see e.g \cite{Abgrall2024a} and the references therein. In \cite{Abgrall2025d}, we had shown the connection between PAMPA and the discontinuous Galerkin method, for the linear hyperbolic problem. Taking advantage of this reinterpretation, we use it to define a family of methods, show how to implement the boundary conditions in a rigorous manner. In addition, we propose a method that complements the bound preserving method developed in \cite{Abgrall2025d} in the sense that it is non oscillatory. A truncation error analysis is provided, it shows that the scheme should be third order accurate for smooth solutions. This is confirmed by numerical experiments. Several numerical examples are presented to show that the scheme is indeed bound preserving and non oscillatory on a wide range on numerical benchmarks.

Paper Structure

This paper contains 14 sections, 80 equations, 7 figures, 1 table.

Table of Contents

  1. Keywords
  2. MSCcodes
  3. Introduction
  4. Robust PAMPA Schemes
  5. Third-order PAMPA Scheme and Its DG Formulation
  6. From DG to PAMPA
  7. Low-order PAMPA scheme
  8. Bound Preservation and Oscillation Elimination
  9. Boundary Conditions
  10. Numerical Examples
  11. Scalar case
  12. Euler Equations of Gas Dynamics
  13. Conclusions and Perspectives
  14. Truncation error analysis

Figures (7)

  • Figure 1: DoFs and normals of triangle $K$.
  • Figure 2: Geometry for the first-order scheme.
  • Figure 3: Example 2: Zalesak problem. Numerical solutions at $t=1$.
  • Figure 4: Example 3: KPP problem. Numerical solutions at $t=1$.
  • Figure 5: Example 4: KT problem. Numerical solutions at $t=1$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Remark 2.1: Other possible schemes
  • Remark 3.1
  • Remark A.1