Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation

• Standard time discretization does not guarantee r^{n+1}>0, and the computed value can also differ significantly from the square-root expression when the time step is large.This conflicts with the continuous definition of r.Source: §1 · p. 2
• The paper addresses an overlooked positivity gap within the original square-root definition, unlike variants that preserve positivity by changing the auxiliary-variable formulation.The introduction identifies exponential SAV and generalized SAV as such alternatives.Source: §1 · p. 3
• The original SAV relation is a simple algebraic link to nonlinear energy, and fixing its sign-indeterminacy at the source strengthens the broader SAV framework without added computational cost.The introduction notes that relaxed variants use the original SAV solution as the starting point for post-processing.Source: §1 · p. 3
• The proposed schemes retain the square-root definition and guarantee a unique positive auxiliary-variable root; numerical experiments confirm positivity, energy stability, and convergence rates.They are proved unconditionally energy-stable and incur the same computational cost as the original SAV method.Source: §1 · p. 3
• For Crank–Nicolson, discretizing the half-time product r^{n+1}r^n preserves the quadratic structure; the cited arithmetic-mean alternatives instead lead to cubic equations without a comparable guarantee.Numerical experiments are reported to confirm the predicted positivity, energy stability, and convergence rates.Source: §1 · p. 3

2 The original SAV method and its sign-indeterminacy

• Both backward Euler and Crank–Nicolson lack a theoretical guarantee that r^{n+1}>0, because the linear scalar equation’s constant term can have either sign.The sign depends on competition between r^n and a perturbation driven by N(φ^n), which depends on φ^n’s spatial structure.Source: §2 · p. 5, §2.3 · p. 8

2.1 A sufficient condition for positivity

• The sharp criterion gives a computable necessary-and-sufficient positivity test from step-n data, while the operator-norm time-step bound is sufficient but can be considerably more pessimistic.For fixed φ^n satisfying the hypotheses, the sufficient bound guarantees positivity for sufficiently small τ.Source: §2.1 · pp. 5, 7
• For the backward Euler scheme, L≥M guarantees r^{n+1}>0 for every τ; when L<M, Proposition 1 requires a time-step restriction.The sufficient condition is not necessary and may be more pessimistic than the sharp threshold.Source: §2.1 · pp. 6–7

2.2 An explicit counterexample

• For α=12, τ=0.1, C=0, and φ0=2, a single backward Euler step destroys positivity, changing initially positive r0=1.5 to a nonpositive value and demonstrating sign failure.Whenever |φ0|>1, the right-hand side is strictly positive, so a finite τ for given α or finite α for given τ triggers sign failure.Source: §2.2 · pp. 7–8

2.3 Structural nature of the sign-indeterminacy

• Directly discretizing r_t creates a linear equation whose indefinite-sign intercept permits sign loss; rewriting the evolution in terms of (r^2)_t yields a convex quadratic with a unique positive root.The quadratic has strictly negative constant term −(r^n)^2, so its structure ensures the positive root regardless of the data.Source: §2.3 · p. 8
• In PDEs, sign failure can involve high-frequency content: even smooth data can trigger it when sufficient amplitude lies at wave numbers whose GL eigenvalues are large relative to τ^-1.A direct ODE-threshold translation using ∥GL∥ can be overly pessimistic because it ignores Fourier-coefficient decay for smooth solutions.Source: §2.2 · p. 8, §2.3 · p. 9

3 Positivity-preserving SAV schemes

• Rewriting the evolution from r_t to (r^2)_t yields a convex quadratic with a strictly negative constant term and guarantees a unique positive root.This structural change resolves sign-indeterminacy without tuning numerical parameters.Source: §3 · p. 9

3.1 Backward Euler scheme

• The backward Euler scheme has a unique solution with r^{n+1}>0 and unconditionally satisfies the energy dissipation law.Its convex quadratic has a leading coefficient of at least 1 and constant term −(r^n)^2 when r^n>0.Source: §3.1 · p. 10

3.2 Crank–Nicolson scheme

• For Crank–Nicolson, the product form r^{n+1}r^n preserves a quadratic equation and guarantees a unique positive root; arithmetic-mean alternatives yield cubics.The product form is consistent with r(t_{n+1/2})^2 and preserves second-order accuracy; the scheme also unconditionally satisfies the energy dissipation law.Source: §3.2 · pp. 11–12
• The reformulation retains the square-root definition and requires two constant-coefficient linear elliptic solves plus one quadratic formula per time step.It can be combined with relaxed or weighted SAV approaches, whose post-processing does not interfere with the positivity guarantee.Source: §3.2 · p. 12

4 Numerical results

• The experiments use Fourier spectral spatial discretization, periodic boundary conditions, and composite trapezoidal integration for Cahn–Hilliard examples.The reported Ginzburg–Landau parameters are ε = 0.05, γ = 4, and C = 0.Source: §4 · pp. 12–13
• Starting from the same positive r_0, the original SAV-BE produces a negative auxiliary variable in the single-step comparison, whereas the proposed scheme preserves positivity.The proposed scalar equation has constant term −(r_0)^2 < 0 and a unique positive root.Source: §4 · p. 13
• At τ = 10^-3, the tested high-frequency mode has λ = 4800 and τλ/(1 + τλ) ≈ 0.8276, illustrating why time-step reduction alone may be impractical for positivity.For a fixed spatial resolution, high-frequency modes keep this factor near one even when τ is reduced to typical phase-field values.Source: §4 · p. 13
• At T = 0.5, SAV-BE exhibits first-order accuracy and SAV-CN second-order accuracy; at comparable steps, SAV-CN profiles are closer to the reference.The convergence orders are reported as consistent with theoretical expectations.Source: §4 · pp. 14–15, Table 1 · p. 14
• Across tested step sizes, both schemes kept r_n positive and discrete energy decayed monotonically; both profiles converged toward the reference as τ decreased.Figures 1 and 2 show the SAV-BE and SAV-CN evolutions, respectively.Source: §4 · p. 14, Figure 1 · p. 14, Figure 2 · p. 15

5 Conclusion

• Reformulating the discrete evolution from r_t to (r^2)_t produces a convex quadratic with strictly negative constant term −(r_n)^2 and a unique positive root, at the original SAV method’s computational cost.The resulting SAV-BE and SAV-CN schemes are proved unconditionally energy-stable.Source: §5 · p. 15
• The product-form r_(n+1)r_n discretization is essential for Crank–Nicolson: arithmetic or squared means yield cubic equations without a comparable positivity guarantee.Source: §5 · p. 15
• The reformulation may combine with relaxed or weighted SAV frameworks, and DIRK stages reduce to BE-type quadratics; multistep methods beyond one step break that structure and lack an unconditional positivity proof.Source: §5 · p. 15
• Rigorous error estimates remain to be developed, while the algebraic reformulation may also apply to SAV discretizations of Hamiltonian and other energy-conserving systems.Source: §5 · p. 16