Compact Finite-Average-Moment Schemes with Single-Step Oscillation Elimination for Hyperbolic Conservation Laws
Abstract
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 ℙ1 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 kth-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.
Related
- Entropy bounded compact stencil Runge-Kutta discontinuous Galerkin method for hyperbolic conservation laws
- Fully discrete multi-entropy-stable compact Runge--Kutta discontinuous Galerkin methods for hyperbolic conservation laws
- Runge--Kutta-Aligned Oscillation Elimination: Restoring Superconvergence for Fully Discrete Shock-Capturing DG Schemes
- Stable Ergodicity for Expanding on Average Partially Hyperbolic Diffeomorphisms with Two-Dimensional Center
- Overcompressive shocks in a 2×2 hyperbolic system of conservation laws
BibTeX
@misc{fan2026compact,
title = {{Compact Finite-Average-Moment Schemes with Single-Step Oscillation Elimination for Hyperbolic Conservation Laws}},
author = {Chuan Fan and Kailiang Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26194},
eprint = {2609.26194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26194},
}
RIS
TY - UNPB
TI - Compact Finite-Average-Moment Schemes with Single-Step Oscillation Elimination for Hyperbolic Conservation Laws
AU - Chuan Fan
AU - Kailiang Wu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26194
UR - https://arxiv.org/abs/2609.26194
AB - 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 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 kth-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.
ER -