Exponential Time Differencing Schemes for a Phase-Field Model of Multicomponent Membranes
This work provides efficient and stable numerical methods for simulating coupled membrane deformation and protein segregation, which is important for computational biophysics.
The authors developed exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes, achieving unconditional energy stability and discrete maximum-bound preservation. Numerical experiments confirmed temporal accuracy and enabled stable 3D simulations of protein-driven pattern formation and membrane deformation.
In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introduce a geometry-adapted operator splitting method based on the localization function, which reformulates the surface OK dynamics into a form suitable for ETD integration. The resulting first- and second-order ETD schemes, combined with finite-difference spatial discretization, are rigorously proved to satisfy a discrete maximum-bound principle and unconditional energy stability. For the coupled system, we construct stabilized ETD schemes in an FFT-based spectral framework, treating stiff linear terms exactly and nonlinear mechanochemical couplings explicitly. A narrow-band implementation further reduces the computational cost by restricting surface calculations to the diffuse membrane region. Numerical experiments confirm the predicted temporal accuracy, maximum-bound preservation, and energy decay for the fixed-membrane OK problem, and demonstrate stable and efficient three-dimensional simulations of protein-driven pattern formation and membrane deformation.