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

Zhiwei Zhang, Shuwang Li, John Lowengrub, Steven M. Wise

TL;DR

A constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn-Hilliard evolution equation rather than the chemical potential is introduced, making CC-MSAV particularly well suited for large-scale simulations of vesicle dynamics.

Abstract

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.

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

TL;DR

A constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn-Hilliard evolution equation rather than the chemical potential is introduced, making CC-MSAV particularly well suited for large-scale simulations of vesicle dynamics.

Abstract

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.
Paper Structure (75 sections, 4 theorems, 103 equations, 7 figures, 5 tables, 1 algorithm)

This paper contains 75 sections, 4 theorems, 103 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Formulation of the Phase-Field Model
  3. Phase-Field Vesicle Model
  4. Governing equations
  5. Free energy functional
  6. Chemical potentials and boundary conditions
  7. Energy dissipation
  8. MSAV Reformulation
  9. Modified energy functional
  10. Variational derivatives and reformulated equations
  11. Auxiliary variable evolution and initial conditions
  12. Energy dissipation of the reformulated system
  13. Numerical Scheme Development
  14. Overview of the Numerical Approach
  15. Temporal Discretization and Stabilization Strategy
  16. ...and 60 more sections

Key Result

Theorem 3.1

Consider the numerical scheme eq:AC_BDF2--eq:Z_BDF2 with the modified discrete energy represents the stabilization correction. Then the following stability property holds for all $n \geq 2$: where $K^n \geq 0$ represents accumulated numerical dissipation arising from the BDF2 second-order differences $\|a^{n+1}-2a^n+a^{n-1}\|^2$ across all discrete variables that are not scaled by $\lambda$(see s

Figures (7)

  • Figure 1: Runtime comparison for proposed CC-MSAV+DCT (blue) versus classical MSAV with sparse direct solver (red) and DCT-preconditioned PCG (green) on grids $N=128$--$4096$. (Top left) Total simulation time for 100 time steps. (Top right) Per-timestep solver cost for Cahn--Hilliard subsystem. (Bottom left) Speedup factors showing 6--15$\times$ acceleration at $N=4096$. (Bottom right) PCG iteration counts (green diamonds) remain mesh-independent at 12--13 iterations; blue line marks zero iterations for direct DCT. All timings in MATLAB R2025b on identical hardware.
  • Figure 2: Spatial convergence for $\phi$ (left), $\psi$ (middle), and combined system (right). Dashed lines show theoretical $O(h^2)$ rate.
  • Figure 3: Elliptical growth validation. Top: Interface evolution (SAV: row 1; NLMG: row 2) at $t = 0, 0.005, 0.01, 0.02$. Bottom: Energy components, arc length, mass, and concentration dynamics. Excellent agreement validates physical accuracy.
  • Figure 4: Elliptical shrinkage validation with $\gamma_{\text{bend}} = 1.0$ at $t = 0, 0.0012, 0.005, 0.02$. Top: Morphology (SAV: row 1; NLMG: row 2). Bottom: Quantitative comparison showing consistent dissipation, conservation, and dynamics.
  • Figure 5: Accuracy comparison for the elliptical growth test. The first row corresponds to the classical MSAV method and the second row corresponds to the proposed CC-MSAV method. The top panels show the interface evolution at $t = 0$, $0.0012$, $0.005$, and $0.02$. The bottom panels contain the energy components and conservation metrics. The results remain visually and quantitatively indistinguishable, which confirms that the efficiency gains reported in Figure \ref{['fig:runtime_comparison']} are obtained without a reduction in accuracy.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Remark 1
  • Remark 2: Selection of stabilization parameters
  • Theorem 3.1: Unconditional Energy Stability
  • proof
  • Remark 3: Unconditional energy stability
  • Theorem 3.2: Fully-Discrete Energy Stability
  • proof
  • Remark 4: Mass conservation
  • Lemma 1: BDF2 Identities
  • Remark 5: Well-posedness of Linear Operators
  • ...and 5 more