标题： 一种新颖且简单的保域框架用于PAMPA方案：一维情况 显示英文标题

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

摘要： PAMPA（Point-Average-Moment PolynomiAl-interpreted）方法由[R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023]提出，结合了双曲守恒律的保守和非保守形式来演化单元平均值和点值。 双曲守恒律的解通常具有不变域，并且确保数值解保持在这个域内是至关重要的，但并不容易实现。 本文提出了一个新颖的框架，用于设计高效的不变域保持（IDP）PAMPA格式。 我们首先分析了原始PAMPA格式中更新后的单元平均值的IDP性质，揭示了单元平均值分解和中点值在保持不变域中的作用。 这一分析强调了仅仅依赖连续通量来保持不变域的难度。 基于这些见解，我们引入了一个简单的IDP限制器用于单元中点值，并提出了一种可证明的IDP PAMPA格式，保证更新后的单元平均值保持不变域，而无需额外的后处理限制器。 这种方法与现有的边界保持PAMPA格式形成对比，后者通常需要额外的凸限制来混合高阶和低阶解。 最值得注意的是，受机器学习中的Softplus和Clipped ReLU函数的启发，我们提出了控制方程的自动IDP重新表述，从而得到一种无条件无需限制器的IDP格式以演化点值。 我们还介绍了抑制虚假振荡的技术，使该格式能够有效地捕捉强激波。 一维问题上的数值实验，包括线性对流方程、Burgers方程、可压缩欧拉方程和MHD方程，展示了所提出的IDP PAMPA格式的准确性和鲁棒性。 显示英文摘要

摘要： The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023], combines conservative and non-conservative formulations of hyperbolic conservation laws to evolve cell averages and point values. Solutions to hyperbolic conservation laws typically have an invariant domain, and ensuring numerical solutions stay within this domain is essential yet nontrivial. This paper presents a novel framework for designing efficient Invariant-Domain-Preserving (IDP) PAMPA schemes. We first analyze the IDP property for updated cell averages in the original PAMPA scheme, revealing the role of cell average decomposition and midpoint values in maintaining the invariant domain. This analysis highlights the difficulty of relying on continuous fluxes alone to preserve the invariant domain. Building on these insights, we introduce a simple IDP limiter for cell midpoint values, and propose a provably IDP PAMPA scheme that guarantees the preservation of the invariant domain for updated cell averages without requiring post-processing limiters. This approach contrasts with existing bound-preserving PAMPA schemes, which often require additional convex limiting to blend high-order and low-order solutions. Most notably, inspired by the Softplus and Clipped ReLU functions from machine learning, we propose an automatic IDP reformulation of the governing equations, resulting in an unconditionally limiter-free IDP scheme for evolving point values. We also introduce techniques to suppress spurious oscillations, enabling the scheme to capture strong shocks effectively. Numerical experiments on 1D problems, including the linear convection equation, Burgers equation, the compressible Euler equations, and MHD equations, demonstrate the accuracy and robustness of the proposed IDP PAMPA scheme.