Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation
Abstract
The scalar auxiliary variable (SAV) method replaces the nonlinear part of the free energy by a positive scalar r(t)=√(ℰN[ϕ]+C)>0, thereby yielding linear, unconditionally energy-stable schemes for gradient flows. At the discrete level, however, the standard backward Euler and Crank--Nicolson discretizations provide no guarantee that the computed rn+1 remains positive, an inconsistency with the continuous definition that contradicts the square-root ansatz and may compromise long-time robustness. Although the SAV method has been widely applied, this subtle but consequential issue has received little attention. We first characterize this failure quantitatively by deriving a sharp criterion and a sufficient condition on the time step size, and construct an explicit counterexample showing that sign loss occurs for parameters of practical relevance. Rather than modifying the definition of r as in existing positivity-preserving variants, we retain the square-root form and reformulate the discrete evolution from rt to (r2)t, which converts the scalar equation into a convex quadratic with a strictly negative constant term, always yielding a unique positive root. For the Crank--Nicolson scheme, the product-form discretization rn+1rn preserves this quadratic structure, while conventional alternatives do not. The resulting schemes incur the same computational cost as the original SAV method and are proved unconditionally energy-stable. Numerical experiments for the Cahn--Hilliard equation confirm the predicted positivity, energy stability, and convergence rates.
Related
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BibTeX
@misc{huang2026positivitypreserving,
title = {{Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation}},
author = {Qiong-Ao Huang and Zhi-Hao Liu and Ying-Wei Wang and Li-Na Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01063},
eprint = {2610.01063},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01063},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation
AU - Qiong-Ao Huang
AU - Zhi-Hao Liu
AU - Ying-Wei Wang
AU - Li-Na Yan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01063
UR - https://arxiv.org/abs/2610.01063
AB - The scalar auxiliary variable (SAV) method replaces the nonlinear part of the free energy by a positive scalar r(t)=√(ℰ__N[ϕ]+C)>0, thereby yielding linear, unconditionally energy-stable schemes for gradient flows. At the discrete level, however, the standard backward Euler and Crank--Nicolson discretizations provide no guarantee that the computed r^n+1 remains positive, an inconsistency with the continuous definition that contradicts the square-root ansatz and may compromise long-time robustness. Although the SAV method has been widely applied, this subtle but consequential issue has received little attention. We first characterize this failure quantitatively by deriving a sharp criterion and a sufficient condition on the time step size, and construct an explicit counterexample showing that sign loss occurs for parameters of practical relevance. Rather than modifying the definition of r as in existing positivity-preserving variants, we retain the square-root form and reformulate the discrete evolution from r_t to (r^2)_t, which converts the scalar equation into a convex quadratic with a strictly negative constant term, always yielding a unique positive root. For the Crank--Nicolson scheme, the product-form discretization r^n+1r^n preserves this quadratic structure, while conventional alternatives do not. The resulting schemes incur the same computational cost as the original SAV method and are proved unconditionally energy-stable. Numerical experiments for the Cahn--Hilliard equation confirm the predicted positivity, energy stability, and convergence rates.
ER -