Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn-Hilliard systems with bounded mass source

TL;DR

This paper proves the stability and convergence of a relaxed scalar auxiliary variable scheme for Cahn-Hilliard systems with bounded mass source, extending SAV methods to non-dissipative systems in applications like image processing and biological modeling.

Contribution

It establishes stability and convergence results for a relaxed SAV scheme applied to Cahn-Hilliard systems with mass sources, a case not covered by previous SAV analyses.

Findings

Proved stability of the relaxed SAV scheme for systems with bounded mass source.

Demonstrated convergence of the scheme in the specified setting.

Applied the scheme to practical problems in image processing and biological systems.

Abstract

The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn-Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg-Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn-Hilliard systems appearing in diblock co-polymer phase separation, tumor growth, image inpainting and segmentation.