A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model

Abstract

Different from the classical phase-field elastic bending model of lipid vesicles that uses a penalty term to conserve volume approximately, in this paper, a new model with accurate volume conservation is first established. Then, for its coupling system with the incompressible flow, we design a highly efficient scheme which is linear and energy stable. More importantly, this scheme is second-order time-accurate and fully-decoupled and it only needs to solve several independent linear equations with constant coefficients at each time step to obtain a numerical solution with second-order time accuracy. The key idea is to introduce two types of nonlocal auxiliary variables, one of which is linearize the nonlinear potential, and the other is used to introduce an ordinary differential equation to deal with the nonlinear coupling terms that satisfy the “zero-energy-contribution” feature. We strictly prove the solvability and unconditional energy stability and conduct numerical simulations in 2D and 3D to demonstrate the accuracy and stability of the scheme numerically. To the best of the author's knowledge, the decoupling method developed in this paper is the first second-order fully-decoupled scheme for the flow-coupled phase-field model.