Efficient second-order physics-preserving schemes for incompressible flows with variable density and viscosity
Abstract
This paper proposes a novel second-order prediction--filter--projection framework for incompressible flows with variable density and viscosity. In this framework, a constrained L2-projection is enforced to unconditionally preserve the prescribed density bounds and discrete mass conservation, while a generalized-Laplacian reformulation combined with the D-CSAV approach is used to discretize the variable-viscosity momentum equation. In particular, a density-weighted second-order time filter is proposed, which eliminates the additional uniform-in-time bound restriction imposed on the time difference quotient of the discrete density in existing stability analyses. The resulting scheme is shown to satisfy an unconditional modified-energy dissipation law, without the viscosity-related restrictions imposed in earlier work. Then, an additional scalar auxiliary variable and its compatible temporal discretization are introduced to construct a variable-step prediction--filter--projection scheme. This scheme is also bound-preserving and mass-conserving, and satisfies an unconditional modified-energy dissipation law. Moreover, the modified energy is much closer to the ``original'' kinetic energy, differing from it only through the auxiliary-variable corrections. Based on the variable-step scheme, a reliable energy-based adaptive time-stepping strategy is developed. Finally, numerical experiments demonstrate the accuracy, physics-preserving properties, and efficiency of the proposed schemes.
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BibTeX
@misc{kuang2026efficient,
title = {{Efficient second-order physics-preserving schemes for incompressible flows with variable density and viscosity}},
author = {Baolin Kuang and Lanxin Sun and Rihui Lan and Hongfei Fu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00834},
eprint = {2611.00834},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00834},
}
RIS
TY - UNPB
TI - Efficient second-order physics-preserving schemes for incompressible flows with variable density and viscosity
AU - Baolin Kuang
AU - Lanxin Sun
AU - Rihui Lan
AU - Hongfei Fu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00834
UR - https://arxiv.org/abs/2611.00834
AB - This paper proposes a novel second-order prediction--filter--projection framework for incompressible flows with variable density and viscosity. In this framework, a constrained L^2-projection is enforced to unconditionally preserve the prescribed density bounds and discrete mass conservation, while a generalized-Laplacian reformulation combined with the D-CSAV approach is used to discretize the variable-viscosity momentum equation. In particular, a density-weighted second-order time filter is proposed, which eliminates the additional uniform-in-time bound restriction imposed on the time difference quotient of the discrete density in existing stability analyses. The resulting scheme is shown to satisfy an unconditional modified-energy dissipation law, without the viscosity-related restrictions imposed in earlier work. Then, an additional scalar auxiliary variable and its compatible temporal discretization are introduced to construct a variable-step prediction--filter--projection scheme. This scheme is also bound-preserving and mass-conserving, and satisfies an unconditional modified-energy dissipation law. Moreover, the modified energy is much closer to the ``original'' kinetic energy, differing from it only through the auxiliary-variable corrections. Based on the variable-step scheme, a reliable energy-based adaptive time-stepping strategy is developed. Finally, numerical experiments demonstrate the accuracy, physics-preserving properties, and efficiency of the proposed schemes.
ER -