Analogous DMP and energy-law results for SWIPDP-L (including coupled settings)

Establish discrete maximum principle and energy-law results for the Symmetric Weighted Interior Penalty Diffusive Projection scheme with Zhang–Shu scaling limiter (SWIPDP-L), analogous to the results available for related discontinuous Galerkin schemes that directly provide such properties even for coupled formulations.

Background

The authors compare their method to schemes for which discrete maximum principles and energy laws are directly provable, including in coupled settings. For SWIPDP-L, the presence of the limiter complicates the analysis.

They explicitly postpone proving analogous guarantees for SWIPDP-L due to technical difficulties related to limiting and energy dissipation.

References

We defer to future work the proof of analogous results for the SWIPDP-L scheme, whose analysis is more involved due to the necessity of employing a limiter for boundedness in Theorem \ref{thm:boundedVhk} and the uncertainty regarding energy dissipation discussed in Remark \ref{rem:limitedEnergyDissipation}.

— 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 2)