The spectral radius of iterative methods for the Cahn–Hilliard equation and its relation to the splitting technique

Abstract

We numerically study the stability of implicit schemes for the Cahn–Hilliard equation. The Cahn–Hilliard equation has an extra limitation for numerical schemes: the total free energy has to be non-increasing with time. One of the most popular remedies for this problem is the splitting technique, when the specific free energy is divided into two parts, one of them is treated explicitly and other one is treated implicitly. We analyse this approach in relation to the spectral radius in the case of Jacobi and Gauss–Seidel methods and show that the linear splitting can lead to the deterioration of a numerical algorithm. We also point out the difference between considering the Cahn–Hilliard equation straightforwardly as the single equation of the 4th order or as the system of two equations of the 2nd order. We propose a simple method to control the spectral radius and increase the stability of iterative methods by adding a stabilizing term, equivalent to adding the artificial time derivative of the chemical potential.