Bound preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme: one-dimensional case

TL;DR

This paper introduces a flexible bound-preserving scheme for numerical solutions of PDEs that blends high-order and first-order methods without explicit reconstructions, ensuring positivity and stability.

Contribution

It develops a novel, explicit blending approach for bound preservation in high-order schemes, applicable to scalar and system PDEs, based on optimal blending parameters derived from a geometric quasi-linearization framework.

Findings

The scheme effectively preserves bounds in scalar and system problems.

Numerical tests demonstrate high accuracy and robustness.

Optimal blending coefficients are explicitly derived.

Abstract

We propose a bound-preserving (BP) Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme by blending third-order and first-order constructions. The originality of the present construction is that it does not need any explicit reconstruction within each element, and therefore the construction is very flexible. The scheme employs a classical blending approach between a first-order BP scheme and a high-order scheme that does not inherently preserve bounds. The proposed BP PAMPA scheme demonstrates effectiveness across a range of problems, from scalar cases to systems such as the Euler equations of gas dynamics. We derive optimal blending parameters for both scalar and system cases, with the latter based on the recent geometric quasi-linearization (GQL) framework of [Wu \& Shu, {\em SIAM Review}, 65 (2023), pp. 1031--1073]. This yields explicit, optimal blending coefficients that ensure positivity and control spurious oscillations in both point values and cell averages. This framework incorporates a convex blending of fluxes and residuals from both high-order and first-order updates, facilitating a rigorous BP property analysis. Sufficient conditions for the BP property are established, ensuring robustness while preserving high-order accuracy. Numerical tests confirm the effectiveness of the BP PAMPA scheme on several challenging problems.