Error analysis of the explicit-invariant energy quadratization (EIEQ) numerical scheme for solving the Allen–Cahn equation

This paper focuses on the error analysis of a first-order, time-discrete scheme for solving the nonlinear Allen–Cahn equation. The discretization of the nonlinear potential is achieved through the EIEQ method, which employs an auxiliary variable to linearize the nonlinear double-well potential effectively. The energy stability of the scheme is demonstrated, along with its decoupled type implementation. Under a set of reasonable assumptions related to boundedness and continuity, an extensive error analysis is performed. This analysis results in the establishment of L2 and H1 error bounds for the numerical solution. Furthermore, a variety of numerical examples are conducted to illustrate the accuracy of the EIEQ scheme, highlighting its effectiveness in addressing complex dynamical systems governed by the Allen–Cahn equation.