gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis

TL;DR

This paper introduces new linear, unconditionally energy-stable numerical schemes for the Cahn-Hilliard equation based on the gPAV method, offering reduced computational complexity and verified stability and accuracy through analysis and experiments.

Contribution

The paper develops first- and second-order schemes using gPAV for the Cahn-Hilliard equation that are computationally more efficient and unconditionally stable, with comprehensive stability and error analysis.

Findings

Schemes are unconditionally energy-stable and linear.

Computational complexity is approximately half of previous methods.

Numerical experiments confirm stability at large time steps.

Abstract

We present several first-order and second-order numerical schemes for the Cahn-Hilliard equation with discrete unconditional energy stability. These schemes stem from the generalized Positive Auxiliary Variable (gPAV) idea, and require only the solution of linear algebraic systems with a constant coefficient matrix. More importantly, the computational complexity (operation count per time step) of these schemes is approximately a half of those of the gPAV and the scalar auxiliary variable (SAV) methods in previous works. We investigate the stability properties of the proposed schemes to establish stability bounds for the field function and the auxiliary variable, and also provide their error analyses. Numerical experiments are presented to verify the theoretical analyses and also demonstrate the stability of the schemes at large time step sizes.