Energy-monotonicity enforcement for the SWIPDP-L limiter at higher polynomial order

Determine a constructive procedure to enforce monotone discrete energy dissipation for the Symmetric Weighted Interior Penalty Diffusive Projection scheme with Zhang–Shu scaling limiter (SWIPDP-L) at polynomial order p > 0 by selecting element-wise scaling factors within the limiter so that the discrete energy at each time step does not exceed the previous-step energy, despite the global and non-convex nature of this constraint.

Background

The paper proves monotone energy dissipation for the unlimited SWIPDP scheme and for the limited scheme only in the lowest-order case (p = 0). For p > 0, the authors observe energy dissipation numerically but do not have a proof for the limiter-augmented method.

They propose an optimization view that selects modified scaling coefficients within the Zhang–Shu limiter to enforce a stepwise energy decrease, but note that this constraint is global and non-convex, making an enforcement mechanism nontrivial.

References

Nevertheless, it remains unclear how to enforce such a constraint, as it constitutes a global condition on a non-convex problem.

— 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 Remark "Limited energy dissipation", Section 4 (Discrete maximum principle)