A computationally optimal relaxed scalar auxiliary variable approach for solving gradient flow systems

Abstract

In this paper, we give a computationally optimal relaxed scalar auxiliary variable (SAV) approach for solving gradient flow systems, in which we only need to solve an elliptic partial differential equation with constant coefficients at each time step. In addition to being applicable to various types of auxiliary variables, there are several advantages of our methods, including: (i) by modifying the optimization procedure in correction of auxiliary variables into a linear programming problems, the original hard-to-solve nonlinear optimization problem arising from previous relaxed SAV approach introduced in Jiang et al. (2022) [28] can be avoided; (ii) the resulting method yields some novel linear unconditionally energy stable schemes, in which backward Euler and Crank–Nicolson formulas are used to discretize the time so that the accuracy can reach the first- and second-order, respectively; (iii) comparing with the baseline SAV approach, the discrete energy in the energy dissipation law is closer to the original energy. Finally, ample numerical results demonstrate the accuracy, efficiency and energy stability of the proposed scheme.