Convergence Analysis for the Invariant Energy Quadratization (IEQ) Schemes for Solving the Cahn–Hilliard and Allen–Cahn Equations with General Nonlinear Potential

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. The well-posedness, unconditional energy stabilities and optimal error estimates of the numerical schemes are proved rigorously. Through the comparisons with some other prevalent schemes for several benchmark numerical examples, we demonstrate the stability and the accuracy of the schemes numerically.