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

TL;DR

This paper introduces an improved, globally continuous PAMPA scheme for hyperbolic problems on unstructured meshes, emphasizing bound preservation, non-oscillatory behavior, and rigorous boundary condition implementation.

Contribution

It develops a new PAMPA variant with boundary condition handling, non-oscillatory properties, and third-order accuracy, validated through numerical experiments.

Findings

The scheme is bound preserving across various benchmarks.

It is non oscillatory and third order accurate for smooth solutions.

Numerical experiments confirm the theoretical properties.

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.