An Efficient Constant-Coefficient MSAV Scheme for Computing Vesicle Growth and Shrinkage

We present a fast, unconditionally energy-stable numerical scheme for simulating vesicle deformation under osmotic pressure using a phase-field approach. The model couples an Allen-Cahn equation for the biomembrane interface with a variable-mobility Cahn-Hilliard equation governing mass exchange across the membrane. Classical approaches, including nonlinear multigrid and Multiple Scalar Auxiliary Variable (MSAV) methods, require iterative solution of variable-coefficient systems at each time step, resulting in substantial computational cost. We introduce a constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn-Hilliard evolution equation rather than the chemical potential. This reformulation yields fully decoupled constant-coefficient elliptic problems solvable via fast discrete cosine transform (DCT), eliminating iterative solvers entirely. The method achieves O(N^2 log N) complexity per time step while preserving unconditional energy stability and discrete mass conservation. Numerical experiments verify second-order temporal and spatial accuracy, mass conservation to relative errors below 5 x 10^-11, and close agreement with nonlinear multigrid benchmarks. On grids with N >= 2048, CC-MSAV achieves 6-15x overall speedup compared to classical MSAV with optimized preconditioning, while the dominant Cahn-Hilliard subsystem is accelerated by up to two orders of magnitude. These efficiency gains, achieved without sacrificing accuracy, make CC-MSAV particularly well suited for large-scale simulations of vesicle dynamics.

💡 Research Summary

The paper introduces a highly efficient, unconditionally energy‑stable numerical scheme for simulating vesicle deformation driven by osmotic pressure, based on a phase‑field model that couples an Allen‑Cahn equation (for the membrane interface) with a variable‑mobility Cahn‑Hilliard equation (for solute concentration). Traditional approaches, including nonlinear multigrid (NLMG) solvers and the Multiple‑Scalar‑Auxiliary‑Variable (MSAV) framework, require the solution of variable‑coefficient linear systems at each time step, which becomes prohibitively expensive for large grids (N ≥ 2048).

The key innovation is the Constant‑Coefficient MSAV (CC‑MSAV) scheme. Instead of adding stabilization to the chemical potentials (as in classical MSAV), the authors embed a linear stabilization term directly into the Cahn‑Hilliard evolution equation. By wrapping the entire osmotic energy term into a single scalar auxiliary variable, the chemical potential ν becomes a purely algebraic function of the fields, containing no spatial derivatives of the unknown ψⁿ⁺¹. Consequently, the Cahn‑Hilliard equation takes the form

 ∂ψ/∂t = ∇·