Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach

TL;DR

This paper develops and analyzes stable, accurate numerical schemes for solving the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potentials using the Invariant Energy Quadratization method.

Contribution

It introduces a new semi-discrete time stepping scheme based on the Invariant Energy Quadratization approach, with proven stability and optimal error estimates for broad classes of nonlinear potentials.

Findings

Proved unconditional energy stability of the schemes.

Established optimal error estimates under mild conditions.

Validated the approach for common nonlinear potentials like double-well and Flory-Huggins.

Abstract

In this paper, we carry out stability and error analyses for two first-order, semi-discrete time stepping schemes, which are based on the newly developed Invariant Energy Quadratization approach, for solving the well-known Cahn-Hilliard and Allen-Cahn equations with general nonlinear bulk potentials. Some reasonable sufficient conditions about boundedness and continuity of the nonlinear functional are given in order to obtain optimal error estimates. These conditions are naturally satisfied by two commonly used nonlinear potentials including the double-well potential and regularized logarithmic Flory-Huggins potential. The well-posedness, unconditional energy stabilities and optimal error estimates of the numerical schemes are proved rigorously.