Energy law for the two-step limiter scheme of Liu et al. (2024)

Determine whether the two-step limiter discontinuous Galerkin scheme in Liu et al. (2024) for the quartic Cahn–Hilliard problem satisfies a discrete energy law, and if so, derive a rigorous proof.

Background

The paper highlights that a recent two-step limiter approach achieves a discrete maximum principle at arbitrary order but notes uncertainty about its energy behavior.

Clarifying whether that method satisfies a discrete energy law would enable more direct comparisons of structure-preserving properties across schemes.

References

Moreover, it is unclear whether the scheme in satisfies an energy law, see for instance Remark \ref{rem:limitedEnergyDissipation} for more details, or to what extent it remains competitive in that regard.

— A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order  (2604.00988 - Gunnarsson et al., 1 Apr 2026) in Subsection "Comparison to other schemes", Section 4 (item 1)