Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model

We consider in this paper gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar auxiliary variable (SAV) approach, and we develop the multiple scalar auxiliary variable (MSAV) approach to deal with these cases. We apply the MSAV approach to the phase-field vesicle membrane (PF-VMEM) model which, in addition to some usual nonlinear terms in the free energy, has two additional penalty terms to enforce the volume and surface area. The MSAV approach enjoys the same computational advantages as the SAV approach but can handle free energies with multiple disparate terms such as the volume and surface area constraints in the PF-VMEM model. The MSAV schemes are unconditional energy stable and second-order accurate in time and lead to decoupled elliptic equations with constant coefficients to solve at each time step. Hence, these schemes are easy to implement and extremely efficient when coupled with an adaptive time stepping. Ample numerical results are presented to validate the stability and accuracy of the MSAV schemes.