A mobility-SAV approach for a Cahn-Hilliard equation with degenerate mobilities

TL;DR

This paper introduces new scalar auxiliary variable schemes for the Cahn-Hilliard equation with degenerate mobilities, improving numerical stability and efficiency in simulating surface diffusion flows.

Contribution

The paper proposes novel first- and second-order SAV schemes that relax mobility instead of energy, offering better accuracy and computational efficiency for degenerate mobility models.

Findings

Schemes are theoretically analyzed in the gradient flow context.

Numerical experiments demonstrate improved accuracy and efficiency.

Relaxation of mobility enhances stability and reduces computational costs.

Abstract

A novel numerical strategy is introduced for computing approximations of solutions to a Cahn-Hilliard model with degenerate mobilities. This model has recently been introduced as a second-order phase-field approximation for surface diffusion flows. Its numerical discretization is challenging due to the degeneracy of the mobilities, which generally requires an implicit treatment to avoid stability issues at the price of increased complexity costs. To mitigate this drawback, we consider new first- and second-order Scalar Auxiliary Variable (SAV) schemes that, differently from existing approaches, focus on the relaxation of the mobility, rather than the Cahn-Hilliard energy. These schemes are introduced and analysed theoretically in the general context of gradient flows and then specialised for the Cahn-Hilliard equation with mobilities. Various numerical experiments are conducted to highlight the advantages of these new schemes in terms of accuracy, effectiveness and computational cost.