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 timestep, resulting in substantial computational cost. We introduce a constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn–Hilliard evolution equation instead of the chemical potential. This reformulation yields fully decoupled constant-coefficient elliptic problems solvable via fast discrete cosine transform (DCT), thereby eliminating iterative solvers entirely. The method achieves $\mathcal{O}(N^2 {\rm log} N)$ complexity per timestep, where N is the number of gridpoints in each spatial dimension, while preserving unconditional energy stability and discrete mass conservation. Numerical experiments verify second-order temporal and spatial accuracy, mass conservation to relative errors below $5\times 10^{-11},$ and close agreement with nonlinear multigrid benchmarks. On grids with $N\ge 2048,$ CC-MSAV achieves a six-to-fifteenfold 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.