Accuracy and Consistency of the Explicit-Invariant Energy Quadratization Approach for the Cahn-Hilliard Equation

Abstract:

      It is well known that the explicit-invariant energy quadratization (EIEQ) approach can generate fully decoupled, linear and unconditionally energy-stable numerical schemes, so it is favored by many researchers. However, the undeniable fact is that the numerical method obtained by EIEQ approach preserves the ``modified" energy law instead of the original energy. This is mainly due to the introduction of some auxiliary variables in EIEQ scheme, and the truncation error will make the auxiliary variables deviate from the original definition in the process of numerical calculation. The primary objective of this paper is to address this gap by providing the accuracy and consistency of the EIEQ method in the context of the Cahn-Hilliard equation. We introduce a relaxation technique for auxiliary variables and construct two numerical schemes based on EIEQ. The analysis results show that the newly constructed schemes are not only unconditionally energy stable, linear and fully decoupled, but also can effectively correct the errors introduced by auxiliary variables and follow the original energy law. Finally, several 2D and 3D numerical examples illustrate the accuracy and efficiency of the newly constructed numerical schemes.