The stabilized-trigonometric scalar auxiliary variable approach for gradient flows and its efficient schemes

We develop a trigonometric scalar auxiliary variable (TSAV) approach for constructing linear, totally decoupled, and energy-stable numerical methods for gradient flows. An auxiliary variable r based on the trigonometric form of the nonlinear potential functional removes the bounded-from-below restriction. By adding a positive constant greater than 1, the positivity preserving property of r can be satisfied. Furthermore, the phase-field variables and auxiliary variable r can be treated in a totally decoupled manner, which simplifies the algorithm. A practical stabilization method is employed to suppress the effect of an explicit nonlinear term. Using our proposed approach, temporally first-order and second-order methods are easily constructed. We prove analytically the discrete energy dissipation laws of the first- and second-order schemes. Furthermore, we propose a multiple TSAV approach for complex systems with multiple components. A comparison of stabilized-SAV (S-SAV) and stabilized-TSAV (S-TSAV) approaches is performed to show their efficiency. Two-dimensional numerical experiments demonstrated the desired accuracy and energy stability.