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.

翻译：我们通过融合三阶和一阶构造，提出了一种保界（BP）的点-平均-矩多项式插值（PAMPA）格式。本构造的独创性在于无需在每个单元内进行显式重构，因此构造非常灵活。该格式采用经典混合方法，将一个一阶保界格式与一个本身不保持界的高阶格式相结合。所提出的保界PAMPA格式在一系列问题上均表现出有效性，从标量情形到欧拉气体动力学方程组等系统情形。我们推导了标量和系统情形下的最优混合参数，后者基于[Wu & Shu, {\em SIAM Review}, 65 (2023), pp. 1031--1073]近期提出的几何拟线性化（GQL）框架。由此得到了显式的最优混合系数，确保了点值和单元平均的正性并抑制了伪振荡。该框架融合了高阶和一阶更新中的通量与残差凸组合，便于进行严格的保界性质分析。我们建立了保界性质的充分条件，在保持高阶精度的同时确保了鲁棒性。数值实验在多个具有挑战性的问题上验证了保界PAMPA格式的有效性。