A structure-preserving discontinuous Galerkin scheme for the Cahn-Hilliard equation including time adaptivity

TL;DR

This paper introduces a new discontinuous Galerkin scheme for the Cahn-Hilliard equation that preserves structural properties, incorporates adaptive time stepping, and demonstrates robustness and efficiency through numerical tests.

Contribution

A novel structure-preserving spatial discretization combined with adaptive time stepping for the Cahn-Hilliard equation, including transport, with a robust preconditioning strategy.

Findings

Scheme accurately preserves mass and energy dissipation.

Method demonstrates robustness in advection-dominated scenarios.

Numerical tests confirm efficiency and stability of the approach.

Abstract

We present a novel spatial discretization for the Cahn-Hilliard equation including transport. The method is given by a mixed discretization for the two elliptic operators, with the phase field and chemical potential discretized in discontinuous Galerkin spaces, and two auxiliary flux variables discretized in a divergence-conforming space. This allows for the use of an upwind-stabilized discretization for the transport term, while still ensuring a consistent treatment of structural properties including mass conservation and energy dissipation. Further, we couple the novel spatial discretization to an adaptive time stepping method in view of the Cahn-Hilliard equation's distinct slow and fast time scale dynamics. The resulting implicit stages are solved with a robust preconditioning strategy, which is derived for our novel spatial discretization based on an existing one for continuous Galerkin based discretizations. Our overall scheme's accuracy, robustness, efficient time adaptivity as well as structure preservation and stability with respect to advection dominated scenarios are demonstrated in a series of numerical tests.