A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order

TL;DR

This paper introduces a structure-preserving discontinuous Galerkin scheme for the Cahn-Hilliard equations that maintains maximum principles and energy dissipation at arbitrary polynomial orders, validated by numerical experiments.

Contribution

The authors develop a novel DG scheme that preserves degeneracy, maximum principle, and energy dissipation without regularization, applicable to any polynomial order.

Findings

Scheme preserves strict degeneracy and maximum principle.

Numerical experiments confirm optimal convergence rates.

Energy dissipation observed for polynomial orders p ≥ 1.

Abstract

We propose a structure-preserving discontinuous Galerkin scheme for the Cahn--Hilliard equations with degenerate mobility based on the Symmetric Weighted Interior Penalty formulation. By evaluating the mobility at cell averages rather than as a piecewise polynomial, the proposed scheme preserves strict degeneracy and yields a coercivity constant that is independent of the mobility, removing the need for regularisation. Moreover, we establish existence of discrete solutions even with degeneracy via a Leray--Schauder fixed-point argument, and show that the scheme satisfies a provable discrete maximum principle at arbitrary polynomial order $p$ when combined with the Zhang--Shu scaling limiter for $p > 0$ and from the scheme alone for $p = 0$. Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for $p \geq 1$ and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order $p+1$ in $L^2$ and validate structure-preserving properties with numerical results.