Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach
This work provides rigorous numerical analysis for a class of phase-field models, but the approach is an incremental extension of existing methods to more general potentials.
The authors developed and analyzed two first-order time-stepping schemes based on the Invariant Energy Quadratization approach for the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potentials, proving unconditional energy stability and optimal error estimates.
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.