Error analysis of second-order IEQ numerical schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation

Abstract

In this article, we derive error estimates for two second-order numerical schemes for solving the viscous Cahn-Hilliard equation with hyperbolic relaxation, one based on the second-order Crank-Nicolson time marching method and the other on the backward differentiation formula. In both schemes, the nonlinear potential is discretized by the Invariant Energy Quadratization (IEQ) method, which employs an auxiliary variable to produce a linear and unconditional energy-stable structure. After assuming some reasonable boundedness and continuity conditions for the nonlinear function, the optimal error estimate is rigorously derived using mathematical induction. Finally, several numerical experiments are carried out to verify the theoretical predictions of the convergence rate and energy stability of the algorithms.