Fully-discrete Spectral-Galerkin scheme with second-order time-accuracy and unconditionally energy stability for the volume-conserved phase-field lipid vesicle model

Abstract

In this work, for the phase-field model of lipid vesicles with the property of accurate volume conservation, we construct an effective fully-discrete numerical scheme, in which, the time marching method is based on a novel splitting type technique, and space is discretized by using the Spectral-Galerkin method. The advantage of this scheme is its high efficiency and ease of implementation. Specifically, although the model is highly nonlinear, just by solving two independent linear biharmonic equations with constant coefficients at each time step, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. The essence of the scheme is to introduce several additional auxiliary variables and use the specially designed ordinary differential equations to reformulate the system. In this way, energy stability can be obtained unconditionally, while avoiding the calculation of variable-coefficient systems. We strictly prove that the energy stability in the fully-discrete form that the scheme holds and give a detailed implementation process. Numerical experiments in 2D and 3D are further carried out to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.