Energy-stable and boundedness preserving numerical schemes for the Cahn-Hilliard equation with degenerate mobility

TL;DR

This paper introduces two novel energy-stable numerical schemes for the Cahn-Hilliard equation with degenerate mobility, ensuring boundedness and stability, and demonstrates their effectiveness through numerical experiments.

Contribution

The paper develops and analyzes two new non-centered schemes for the Cahn-Hilliard equation with degenerate mobility, proving their energy stability and approximate maximum principle preservation.

Findings

Both schemes are energy stable.

Solutions stay within bounds approximately.

Numerical results confirm accuracy and stability.

Abstract

Two new numerical schemes to approximate the Cahn-Hilliard equation with degenerate mobility (between stable values 0 and 1) are presented, by using two different non-centered approximation of the mobility. We prove that both schemes are energy stable and preserve the maximum principle approximately, i.e. the amount of the solution being outside of the interval [0,1] goes to zero in terms of a truncation parameter. Additionally, we present several numerical results in order to show the accuracy and the well behavior of the proposed schemes, comparing both schemes and the corresponding centered scheme.