Error analysis of full-discrete invariant energy quadratization schemes for the Cahn-Hilliard type equation

Abstract

In this paper, we present the error analysis for a fully discrete scheme of the Cahn-Hilliard type equation. The numerical schedule is developed by first transforming the Cahn-Hilliard type equation into an equivalent form using the invariant energy quadratization (IEQ) technique. Then the equivalent form is discretized by using the linear-implicit Crank-Nicolson method for the time variable and the Fourier pseudo-spectral method for the spatial variables. The resulted full-discrete scheme is linear and unconditionally energy stable, which makes numerical simulation easy to implement. By constructing an appropriate interpolation equation, the uniform boundedness of the numerical solution is obtained theoretically. Then, we prove that the numerical solutions converge with the order $\mathcal{O}((\delta t{)}^2+h^m)$, where $\delta t$ is the temporal step and $h$ is the spatial step with $m$ the regularity of exact solution. Several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the proposed linear scheme. The numerical strategies and analytical tools developed in this paper could be readily applied to study other phase field models or gradient flow problems.