A Novel and Simple Invariant-Domain-Preserving Framework for PAMPA Scheme: 1D Case

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method was intro- duced in [R. Abgrall, Commun. Appl. Math. Comput., 5 (2023) pp. 370--402], as an innovative approach effectively combining the conservative and nonconservative formulations of a hyperbolic system of conservation laws to evolve cell averages and point values. Solutions to hyperbolic con- servation laws typically admit an invariant domain, and preserving numerical solutions within this domain is essential yet nontrivial. In this paper, we propose a novel framework for designing effi- cient Invariant-Domain-Preserving (IDP) PAMPA schemes. We first provide a rigorous theoretical analysis of the IDP property for the updated cell averages in the original PAMPA scheme, revealing the critical roles of cell average decomposition and the enforcement of midpoint values within the invariant domain. This analysis also reveals the challenges of relying solely on single-state fluxes at cell interfaces (as opposed to two-state numerical fluxes) to ensure the updated cell averages within the invariant domain. Building on these insights, we introduce a simple IDP limiter for cell midpoint values and construct a provably IDP PAMPA scheme that always maintains the IDP property for updated cell averages by proof without the need for additional postprocessing limiters. This approach contrasts with existing bound-preserving PAMPA schemes, which typically require extra convex lim- iting to blend high-order and low-order schemes. Most notably, inspired by the Softplus and Clipped ReLU functions from machine learning, we propose a novel, automatic IDP reformulation of the governing equations. Thanks to this new formulation, we design an unconditionally limiter-free IDP scheme for evolving point values. We also introduce novel techniques to suppress spurious oscillations in the IDP PAMPA scheme, allowing for effective capture of strong shocks. Numerical experiments in one dimension, including tests on the linear convection equation, Burgers' equation, shallow water equations, the compressible Euler equations, and MHD equations, demonstrate the accuracy and robustness of the proposed IDP PAMPA scheme.