A compact high-order and positivity-preserving active flux method for compressible Navier--Stokes equations on Cartesian meshes
Abstract
This paper develops a compact fourth-order positivity-preserving active flux (AF) method for the one-and two-dimensional compressible Navier-Stokes equations on Cartesian meshes. The method retains the cell averages and shared point values of the standard third-order AF method as its degrees of freedom. To avoid the order reduction that can arise when diffusion is discretized using operators from the standard third-order AF method, while maintaining compactness, the divergence of the viscous flux is discretized directly using compact fourth-order operators. For the inviscid part, incorporating a downwind point value into the biased stencil yields fourth-order accuracy. A monolithic flux limiting blends high-order total numerical fluxes with loworder positivity-preserving counterparts, treating the inviscid and viscous fluxes jointly while maintaining local conservation. Together with a scaling limiter for point values, this procedure preserves density and pressure positivity for both cell averages and point values. Numerical experiments demonstrate fourth-order convergence, positivity preservation, and accurate resolution of shocks and viscous flow structures. For the two-dimensional viscous shock tube, a comparison with a discontinuous Galerkin method shows improved computational efficiency in resolving complex interaction of shock waves and boundary layers.
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BibTeX
@misc{duan2026compact,
title = {{A compact high-order and positivity-preserving active flux method for compressible Navier--Stokes equations on Cartesian meshes}},
author = {Junming Duan and Wasilij Barsukow and Christian Klingenberg},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29081},
eprint = {2609.29081},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05794424v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A compact high-order and positivity-preserving active flux method for compressible Navier--Stokes equations on Cartesian meshes
AU - Junming Duan
AU - Wasilij Barsukow
AU - Christian Klingenberg
PY - 2026
DA - 2026/09/24
PB - HAL
DO - 10.48550/arxiv.2609.29081
UR - https://hal.science/hal-05794424v1
AB - This paper develops a compact fourth-order positivity-preserving active flux (AF) method for the one-and two-dimensional compressible Navier-Stokes equations on Cartesian meshes. The method retains the cell averages and shared point values of the standard third-order AF method as its degrees of freedom. To avoid the order reduction that can arise when diffusion is discretized using operators from the standard third-order AF method, while maintaining compactness, the divergence of the viscous flux is discretized directly using compact fourth-order operators. For the inviscid part, incorporating a downwind point value into the biased stencil yields fourth-order accuracy. A monolithic flux limiting blends high-order total numerical fluxes with loworder positivity-preserving counterparts, treating the inviscid and viscous fluxes jointly while maintaining local conservation. Together with a scaling limiter for point values, this procedure preserves density and pressure positivity for both cell averages and point values. Numerical experiments demonstrate fourth-order convergence, positivity preservation, and accurate resolution of shocks and viscous flow structures. For the two-dimensional viscous shock tube, a comparison with a discontinuous Galerkin method shows improved computational efficiency in resolving complex interaction of shock waves and boundary layers.
ER -