紧致有限平均矩格式：带单步振荡消除的双曲守恒律方法

中文摘要 AI 辅助

高阶激波捕捉格式常因宽模板、逐阶段非线性权重和昂贵的局部特征分解而面临严重的计算瓶颈。为解决这一问题，我们针对双曲守恒律提出了一种紧致有限平均矩（FAM）框架，该框架将形式空间阶数与所演化的局部自由度数量完全解耦。通过仅演化一个 $\mathbb{P}^1$ 矩状态（即单元平均值和缩放的一阶矩），我们借助紧致、多项式精确的\emph{线性}矩重构实现了高达六阶的精度（$k=3, 4, 5, 6$）。一个关键的算法创新是将非线性稳定性整合到单一的、无导数的振荡消除（OE）过程中，该过程仅在最终龙格-库塔阶段之后应用。此OE过程精确保持单元平均值，同时对一阶矩应用显式指数修正，避免了逐阶段的非线性权重或限制。线性主干的傅里叶分析表明其具有$k$阶精度、强伪模态阻尼以及$(k+1)$阶单元平均超收敛性。对标量方程和欧拉方程的大量数值实验验证了预期的光滑精度和稳健、无振荡的激波分辨率。值得注意的是，通过在守恒变量中直接进行分量重构并简化稳定性处理，FAM格式相比多分辨率加权本质无振荡（MR-WENO）、统一模板Hermite WENO（HWENO-U）和OE-HWENO方法，实现了显著更低的完整运行墙钟时间。

英文摘要

High-order shock-capturing schemes often face severe computational bottlenecks due to wide stencils, stagewise nonlinear weights, and expensive local characteristic decompositions. To address this, we propose a compact finite-average-moment (FAM) framework for hyperbolic conservation laws that completely decouples the formal spatial order from the number of evolved local degrees of freedom. By evolving only a $\mathbb{P}^1$ moment state (i.e., cell averages and scaled first-order moments), we achieve up to sixth-order accuracy ($k=3, 4, 5, 6$) via compact, polynomially exact \emph{linear} moment reconstructions. A key algorithmic innovation is consolidating the nonlinear stabilization into a single, derivative-free oscillation-elimination (OE) procedure applied only after the final Runge--Kutta stage. This OE procedure exactly preserves cell averages while applying an explicit exponential correction to the first-order moments, avoiding repeated stagewise nonlinear weights or limiting. Fourier analysis of the linear backbone demonstrates $k$th-order accuracy, strong spurious mode damping, and $(k+1)$th-order cell-average superconvergence. Extensive numerical experiments on scalar laws and the Euler equations verify the expected smooth accuracy and robust, nonoscillatory shock resolution. Notably, by performing componentwise reconstruction directly in conservative variables and streamlining stabilization, the FAM schemes deliver significantly lower complete-run wall-clock times compared to the multi-resolution weighted essentially non-oscillatory (MR-WENO), unified-stencil Hermite WENO (HWENO-U), and OE-HWENO methods.

发表机构

• Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
• Southern University of Science and Technology（南方科技大学）

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