Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation
The scalar auxiliary variable (SAV) method replaces the nonlinear part of the free energy by a positive scalar $r(t)=\sqrt{\mathcal{E}_{_\mathcal{N}}[ϕ]+C}>0$, 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 $r^{n+1}r^{n}$ 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.