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

TL;DR

This work addresses stability and convergence of relaxed scalar auxiliary variable (RSAV) schemes for Cahn-Hilliard systems with mass sources that break gradient-flow dissipation. By introducing an auxiliary scalar q and a relaxed update, the authors obtain a linear, unconditionally stable time discretization with a modified energy $G_\varepsilon(\varphi,q)$ and prove uniform energy bounds and convergence to a weak solution as $\tau \to 0$, under bounded sources and quartic growth of the potential. They establish stability via a key estimate relating the auxiliary variable to the cubic nonlinearity and analyze two RSAV relaxations for convergence. Numerical experiments in fully discrete finite elements and several applications—diblock copolymer microphase separation, image segmentation, image inpainting, and tumor growth—confirm stability and show RSAV performs comparably or slightly better than SAV in these non-dissipative contexts. Overall, the results extend the SAV framework to Cahn-Hilliard models with mass sources, enabling robust simulations in materials science and image processing where a dissipative energy structure is absent or altered.

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.

Figures (6)

• Figure 1: A plot of the discrete Ginzburg--Landau energy $E = \frac{1}{2} \mathbb{S} \bm{\varphi}^n_h \cdot \bm{\varphi}^n_h + Q(\bm{\varphi}_h^n)^2 - C_0$ for the numerical solutions approximating the three analytical test solutions.
• Figure 2: Switching behavior observed for the optimal values of $\zeta_n^\tau$ computed via \ref{['RSAV:opt']} for various values of $\eta$ and $M$.
• Figure 3: Simulation of diblock copolymer dynamics with \ref{['CHO:diss']}. Top row (subfigures (a), (b), (c), (d)) display the evolution starting with initial condition $\varphi^0 = -0.5 + U[0,0.2]$. Bottom row (subfigures (e), (f), (g), (h)) display the evolution starting with initial condition $\varphi^0 = -0.1 + U[0,0.2]$.
• Figure 4: Image segmentation with \ref{['dis:Seg']}. Segmentation of a cow image (top row) and a blood vessel image (bottom row) with \ref{['dis:Seg']}. The first three subfigures ((a), (b), (c) and (e), (f), (g)) of each row display the 1/2-level set of $\varphi^n$ (colored red) overlayed with the original image at $n = 1000$, $n = 3000$ and $n = 10000$, respectively. The last subfigure ((d) and (h)) of each row display the discrete solution $\varphi^n$ at $n = 10000$.
• Figure 5: Inpainting of a double stripe binary image with \ref{['dis:Inpaint']}.

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2: Unique solvability
• Remark 2.3
• Remark 2.4
• Theorem 2.1: Stability
• proof
• Remark 2.5: Sharpness of \ref{['ass:Ff']}
• Remark 2.6
• Theorem 3.1: Convergence
• proof