Unconditionally energy stable discontinuous Galerkin schemes for the Cahn-Hilliard equation

TL;DR

This paper develops unconditionally energy stable discontinuous Galerkin schemes for the Cahn-Hilliard equation by combining mixed DG spatial discretization with the IEQ approach for time, ensuring energy dissipation and computational efficiency.

Contribution

It introduces a novel IEQ-DG scheme that guarantees unconditional energy stability and efficient solvability for the Cahn-Hilliard equation.

Findings

The IEQ-DG scheme is unconditionally energy dissipative.

Numerical examples confirm the scheme's efficiency and accuracy.

The method preserves key solution properties in 1D and 2D cases.

Abstract

In this paper, we introduce novel discontinuous Galerkin (DG) schemes for the Cahn-Hilliard equation, which arises in many applications. The method is designed by integrating the mixed DG method for the spatial discretization with the \emph{Invariant Energy Quadratization} (IEQ) approach for the time discretization. Coupled with a spatial projection, the resulting IEQ-DG schemes are shown to be unconditionally energy dissipative, and can be efficiently solved without resorting to any iteration method. Both one and two dimensional numerical examples are provided to verify the theoretical results, and demonstrate the good performance of IEQ-DG in terms of efficiency, accuracy, and preservation of the desired solution properties.