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

Abstract

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 $L^2$ and $H^1$ 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.